#7682062

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

4987 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589258);