#7656914

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

4925 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)(9,10)(11,14)(12,16)(13,15)(17,18)(19,30)(20,32)(21,27)(22,25)(23,28)(24,29)(26,31);(1,2)(3,7)(4,8)(5,6)(9,11,13,16)(10,14,15,12)(17,19,24,22)(18,25,29,30)(20,21,31,28)(23,26,27,32);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,26)(19,21)(22,28)(23,30)(24,31)(25,27)(29,32);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,14)(15,16)(17,21,24,28)(18,27,29,23)(19,31,22,20)(25,32,30,26);(1,3,5,4)(2,7,6,8)(17,22,24,19)(18,25,29,30)(20,28,31,21)(23,26,27,32);(1,4,5,3)(2,8,6,7)(17,23,24,27)(18,28,29,21)(19,26,22,32)(20,30,31,25)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=g,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, 7564110);