#8712714

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

4826 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8619910);