#8757899

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

4948 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)(17,18,24,30)(19,27,21,25)(20,29,23,26)(22,28,32,31);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,13)(14,16)(17,19,24,21)(18,25,30,27)(20,31,23,28)(22,29,32,26);(1,2)(3,5)(4,6)(7,8)(10,13)(14,15)(17,20,24,23)(18,26,30,29)(19,31,21,28)(22,27,32,25);(9,10)(11,14)(12,13)(15,16)(17,21,24,19)(18,27,30,25)(20,31,23,28)(22,29,32,26);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,22,24,32)(18,28,30,31)(19,26,21,29)(20,25,23,27);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,23,24,20)(18,29,30,26)(19,31,21,28)(22,27,32,25)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=g,f^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, 8665095);