#7645330

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

5017 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)(18,24)(22,26)(25,27)(29,32);(1,2,5,6)(3,7,4,8)(10,13)(14,16)(17,18,23,24)(19,27,21,25)(20,26,30,22)(28,29,31,32);(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,25,24,27)(20,28,30,31)(22,29,26,32);(9,11)(10,14)(12,15)(13,16)(17,20)(18,26)(19,28)(21,31)(22,24)(23,30)(25,32)(27,29);(1,3,5,4)(2,7,6,8)(17,21,23,19)(18,25,24,27)(20,28,30,31)(22,32,26,29);(1,4,5,3)(2,8,6,7)(17,22,23,26)(18,20,24,30)(19,29,21,32)(25,28,27,31)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,e^2=g,f^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, 7552526);