#7733805

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

4964 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7641001);