#7971696

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

5113 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)(3,7)(4,5)(6,8)(9,10,13,12)(11,15,16,14)(17,18,22,23)(19,25,26,21)(20,28,29,24)(27,31,32,30);(9,11,13,16)(10,14,12,15)(17,19,22,26)(18,21,23,25)(20,27,29,32)(24,31,28,30);(2,5)(6,7)(9,12,13,10)(11,15,16,14)(18,23)(19,26)(20,29)(30,31);(1,2)(3,6)(4,5)(7,8)(17,18)(19,21)(20,24)(22,23)(25,26)(27,31)(28,29)(30,32);(1,3)(2,6)(4,8)(5,7)(17,20)(18,24)(19,27)(21,31)(22,29)(23,28)(25,30)(26,32);(17,21)(18,19)(20,30)(22,25)(23,26)(24,32)(27,28)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7878892);