#8508916

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

4717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416112);