#7733726

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

4936 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7640922);