#135946

order	512
nilpotency class	4
isoclinism	isoclinic class of #2196
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

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

normal subgroups only

This group has 303 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:

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

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,e^b=egi,d^c=dg,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287453, #72343176, #72343184
maximal normal subgroup of	#72287453, #72343176, #72343184

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43142);