#135947

order	512
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

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	#8851
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 299 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287454, #72343184, #72343185
maximal normal subgroup of	#72287454, #72343184, #72343185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43143);