#7642683

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

4988 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)(6,8)(10,13)(11,15)(20,27)(24,29)(25,31)(30,32);(1,2)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32);(9,10,12,13)(11,14,15,16)(18,22)(23,28)(24,29)(30,32);(9,11)(10,14)(12,15)(13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32);(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32);(1,3,4,7)(2,6,5,8)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549879);