#7658859

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

5628 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18)(19,26)(20,25)(21,27)(22,23)(24,29)(28,30)(31,32);(2,3)(6,7)(11,16)(13,15)(18,23)(20,30)(24,26)(27,32);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,31)(30,32);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(17,20,22,30)(18,25,23,28)(19,27,29,32)(21,24,31,26);(17,19)(18,26)(20,27)(21,25)(22,29)(23,24)(28,31)(30,32);(17,21)(18,27)(19,28)(20,24)(22,31)(23,32)(25,29)(26,30)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566055);