#7642015

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

5442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

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, 7549211);