#7074919

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1711 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fh,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,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	#72257326, #72257398, #72264310
maximal normal subgroup of	#72257326, #72257398, #72264310

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6982115);