#7715789

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

4665 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:

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

Polycyclic presentation:

2^9|d^2=gh,f^2=g,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, 7622985);