#7733452

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

4877 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=g,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, 7640648);