#7651230

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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4796 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)(3,7)(9,10,12,14)(11,15,16,13)(17,18,23,21)(19,27,28,24)(20,25,29,30)(22,26,32,31);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,22)(21,27)(23,28)(25,31)(26,30)(29,32);(1,2,4,5)(3,6,7,8)(19,28)(22,32)(24,27)(26,31);(1,3)(2,6)(4,7)(5,8)(17,20)(18,25)(19,22)(21,30)(23,29)(24,31)(26,27)(28,32);(17,21,23,18)(19,27,28,24)(20,25,29,30)(22,31,32,26);(17,22)(18,26)(19,20)(21,31)(23,32)(24,30)(25,27)(28,29)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7558426);