#33741

order	256
nilpotency class	4
isoclinism	isoclinic class of #33607
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	C₂³	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

242 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 103 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,16,22,35)(12,25,29,54)(13,30,32,58)(14,33,31,19)(17,38,40,27)(18,41,42,26)(21,43,45,65)(23,50,48,71)(24,51,49,37)(28,57,56,53)(34,59,60,46)(36,63,62,44)(39,55,52,64)(47,70,69,67)(61,68,66,72);(2,5)(6,8)(9,11,15,22)(10,16,20,35)(12,21,29,45)(13,23,32,48)(14,24,31,49)(17,34,40,60)(18,36,42,62)(19,37,33,51)(25,43,54,65)(26,44,41,63)(27,46,38,59)(28,47,56,69)(30,50,58,71)(39,61,52,66)(53,67,57,70)(55,68,64,72);(9,12,26,55,14,28,32,38,15,29,41,64,31,56,13,27)(10,17,30,53,19,39,42,54,20,40,58,57,33,52,18,25)(11,21,44,68,24,47,48,59,22,45,63,72,49,69,23,46)(16,34,50,67,37,61,62,65,35,60,71,70,51,66,36,43);(1,3,4,7)(2,6,5,8)(11,22)(16,35)(21,45)(23,48)(24,49)(34,60)(36,62)(37,51)(43,65)(44,63)(46,59)(47,69)(50,71)(61,66)(67,70)(68,72)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#141375, #141378, #150165, …
maximal normal subgroup of	#141375, #141378, #150165, …
maximal quotient of	#494206, #495301, #495311, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26729);