#7590563

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=g,d^2=i,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497759);