#7680310

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

5372 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)(6,7)(9,10,13,15)(11,12,16,14)(18,23)(21,25)(24,30)(27,31);(9,11)(10,14)(12,15)(13,16)(17,18)(19,24)(20,21)(22,23)(25,29)(26,27)(28,30)(31,32);(1,2,4,5)(3,7,8,6)(9,12,13,14)(10,11,15,16)(18,23)(20,29)(24,30)(26,32);(1,3,4,8)(2,6,5,7)(9,11)(10,12)(13,16)(14,15)(17,19,22,28)(18,24,23,30)(20,26,29,32)(21,27,25,31);(17,20)(18,25)(19,26)(21,23)(22,29)(24,31)(27,30)(28,32);(17,21,22,25)(18,20,23,29)(19,27,28,31)(24,26,30,32)

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=gh,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7587506);