#8799481

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

5013 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=ghi,e^2=gi,f^2=h,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, 8706677);