#9875501

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

4855 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=gi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9782697);