#7857098

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

5193 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7764294);