#134631

order	512
nilpotency class	4
isoclinism	isoclinic class of #124690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1548
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8590
prime radical	C₂³	#413
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

267 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=h,e^2=hi,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fgh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265701, #72265719, #72265723
maximal normal subgroup of	#72265701, #72265719, #72265723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41827);