#7786309

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

5002 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)(11,12)(13,14)(15,16)(17,18,23,22)(19,27,28,24)(20,25,29,30)(21,31,32,26);(1,2)(3,7)(4,8)(5,6)(17,19)(18,24)(20,21)(22,27)(23,28)(25,26)(29,32)(30,31);(10,14)(11,15)(18,22)(19,28)(20,29)(26,31);(1,3,5,4)(2,7,6,8)(9,11,13,15)(10,12,14,16)(17,20)(18,25)(19,21)(22,30)(23,29)(24,26)(27,31)(28,32);(9,12)(10,15)(11,14)(13,16)(17,21)(18,26)(19,29)(20,28)(22,31)(23,32)(24,30)(25,27);(1,4,5,3)(2,8,6,7)(17,22,23,18)(19,27,28,24)(20,30,29,25)(21,31,32,26)

Polycyclic presentation:

2^9|a^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7693505);