#7729312

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

4771 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=i,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, 7636508);