#10096791

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

4891 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=g,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, 10003987);