#7710473

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

4811 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)(4,7)(10,14)(15,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,31)(27,32)(28,30);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,19,23,22)(18,20,24,26)(21,27,31,30)(25,28,29,32);(2,6)(3,8)(18,24)(20,26)(25,29)(28,32);(9,11,13,12)(10,15,14,16)(17,20,23,26)(18,19,24,22)(21,28,31,32)(25,27,29,30);(1,3,5,8)(2,7,6,4)(9,11,13,12)(10,15,14,16)(17,21,23,31)(18,25,24,29)(19,27,22,30)(20,28,26,32);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,16,14,15)(17,22,23,19)(18,26,24,20)(21,30,31,27)(25,32,29,28)

Polycyclic presentation:

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

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, 7617669);