#7735715

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

4910 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=hi,e^2=h,f^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, 7642911);