#8757848

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

4831 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)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,13)(14,16)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31);(1,2)(3,5)(4,6)(7,8)(10,13)(14,15)(18,20)(21,30)(23,25)(26,29);(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8665044);