#7645317

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

5128 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7552513);