#8699109

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

4793 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8606305);