#10089147

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

4964 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)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32);(1,3,5,7)(2,6,8,4)(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32);(1,3)(2,4)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(18,20)(19,26)(25,28)(27,32);(1,3,5,7)(2,6,8,4)(9,10)(11,13)(12,15)(14,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31);(1,4,5,6)(2,7,8,3)(9,10)(11,13)(12,15)(14,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

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

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, 9996343);