#8621935

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

4925 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)(3,7)(4,6)(5,8)(9,10,14,13)(11,12,16,15)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32);(1,3,6,8)(2,7,4,5)(9,11)(10,15)(12,13)(14,16)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32);(2,4)(3,8)(9,12,14,15)(10,11,13,16)(19,24)(21,29)(27,30)(28,32);(3,8)(5,7)(10,13)(11,16)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32);(1,4,6,2)(3,7,8,5)(9,12)(10,11)(13,16)(14,15)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27);(1,5,6,7)(2,8,4,3)(9,13,14,10)(11,12,16,15)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529131);