#7145448

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,22,26)(19,24,28,31)(20,25,21,30)(23,29,32,27);(2,5)(3,7)(10,13)(11,15);(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,31)(26,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,32)(26,31);(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,31,32,24)

Polycyclic presentation:

2^9|a^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285313, #72313826, #72313934
maximal normal subgroup of	#72285313, #72313826, #72313934

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7052644);