#7721860

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

4950 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7629056);