#6990469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25731	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1697 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6897665);