#8456988

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8364184);