#7714887

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

4758 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622083);