#7719930

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

4821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

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, 7627126);