#7729092

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

4942 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7636288);