#7733366

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

5842 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)(11,14)(12,15)(13,16)(17,18)(19,22)(20,25)(21,29)(23,24)(26,30)(27,31)(28,32);(1,2)(3,7)(4,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,27)(18,22,24,31)(20,21,28,30)(25,29,32,26);(2,5)(6,7)(9,11,12,16)(10,14,15,13)(18,24)(19,27)(21,30)(25,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,23,28)(18,25,24,32)(19,21,27,30)(22,29,31,26);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,21)(18,26)(19,28)(20,27)(22,25)(23,30)(24,29)(31,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,22)(18,19)(20,29)(21,25)(23,31)(24,27)(26,28)(30,32)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^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, 7640562);