#7722689

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

4651 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=gh,f^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, 7629885);