#7782739

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7689935);