#7820136

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

4772 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)(19,28)(20,26)(21,22)(23,25)(24,29)(27,31)(30,32);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,24)(20,21)(22,32)(23,29)(25,28)(26,27)(30,31);(2,5)(6,8)(18,25)(19,29)(20,30)(22,27);(3,7)(6,8)(9,10,13,12)(11,14,16,15)(17,20,23,30)(18,26,25,32)(19,21,29,31)(22,24,27,28);(1,3)(2,6)(4,7)(5,8)(9,10,13,12)(11,14,16,15)(17,21)(18,27)(19,20)(22,25)(23,31)(24,26)(28,32)(29,30);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,15,16,14)(17,22,23,27)(18,21,25,31)(19,26,29,32)(20,24,30,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7727332);