#7726040

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

4652 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=gi,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, 7633236);