#33687

order	256
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	Aut(G) order 8192
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

260 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 101 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)(9,10,13,18)(11,21,23,16)(12,26,27,62)(14,32,30,61)(15,35,31,20)(17,40,41,28)(19,44,43,34)(22,50,51,60)(24,56,54,65)(25,39,55,49)(29,63,36,45)(33,46,64,42)(37,52,47,69)(38,58,48,70)(53,67,59,71)(57,66,72,68);(2,5)(6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,37)(18,21)(19,38)(20,39)(26,60)(27,51)(28,52)(29,53)(30,54)(31,55)(32,65)(33,57)(34,58)(35,49)(36,59)(40,69)(41,47)(42,66)(43,48)(44,70)(45,67)(46,68)(50,62)(56,61)(63,71)(64,72);(9,12,13,27)(10,17,18,41)(11,22,23,51)(14,33,30,64)(15,36,31,29)(16,37,21,47)(19,45,43,63)(20,46,35,42)(24,57,54,72)(25,59,55,53)(26,61,62,32)(28,44,40,34)(38,67,48,71)(39,68,49,66)(50,65,60,56)(52,70,69,58);(1,3,4,7)(2,6,5,8)(11,23)(16,21)(22,51)(24,54)(25,55)(37,47)(38,48)(39,49)(50,60)(52,69)(53,59)(56,65)(57,72)(58,70)(66,68)(67,71)

Polycyclic presentation:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#33687, #7623095, #7623104, …
maximal normal subgroup of	C₂×#33687, #7623095, #7623104, …
maximal quotient of	#494036, #494973, #494981, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26675);