#8754911

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

4778 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662107);