#10088416

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

5027 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=g,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, 9995612);