#33768

order	256
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

276 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,7)(4,6)(5,8)(9,10,15,20)(11,16,22,37)(12,25,28,59)(13,29,33,58)(14,34,31,19)(17,40,42,26)(18,43,45,32)(21,47,48,70)(23,53,51,69)(24,56,52,39)(27,60,35,44)(30,46,61,41)(36,62,63,49)(38,66,65,55)(50,71,57,67)(54,68,72,64);(1,3)(2,5)(4,8)(6,7)(9,11,15,22)(10,16,20,37)(12,21,28,48)(13,23,33,51)(14,24,31,52)(17,36,42,63)(18,38,45,65)(19,39,34,56)(25,47,59,70)(26,49,40,62)(27,50,35,57)(29,53,58,69)(30,54,61,72)(32,55,43,66)(41,64,46,68)(44,67,60,71);(3,8)(5,7)(9,12,15,28)(10,17,20,42)(11,21,22,48)(13,30,33,61)(14,35,31,27)(16,36,37,63)(18,44,45,60)(19,46,34,41)(23,54,51,72)(24,57,52,50)(25,58,59,29)(26,43,40,32)(38,67,65,71)(39,68,56,64)(47,69,70,53)(49,66,62,55);(2,6)(5,7)(11,22)(16,37)(21,48)(23,51)(24,52)(36,63)(38,65)(39,56)(47,70)(49,62)(50,57)(53,69)(54,72)(55,66)(64,68)(67,71)

Polycyclic presentation:

2^8|a^2=h,b^2=h,c^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ce,d^b=dh,f^c=fg,g^c=gh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#142619, #142623, #142699, …
maximal normal subgroup of	#142619, #142623, #142699, …
maximal quotient of	#496271, #496678, #496825, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26756);