#7657894

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

5631 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565090);