#136831

order	512
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8965
prime radical	#3018	C₄
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

495 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=egi,g^a=gi,e^b=egi,d^c=dg,e^c=ei,f^c=fg,e^d=ei,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343176, #72343184, #72343189
maximal normal subgroup of	#72343176, #72343184, #72343189

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44027);