#7782116

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

4582 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7689312);