#8772656

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

4612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,14,18)(11,12,16,15)(13,17,22,24)(19,20,23,21)(26,29)(31,33)(32,37)(38,39);(1,2,4,6)(3,5,8,7)(9,11,14,16)(10,15,18,12)(13,19,22,23)(17,21,24,20)(26,29)(28,36)(31,33)(34,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,15)(10,16,18,11)(13,21,22,20)(17,19,24,23)(25,26,30,29)(27,33,35,31)(28,37,36,32)(34,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,22)(23,24)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,28,30,36)(26,32,29,37)(27,34,35,40)(31,38,33,39);(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,29,30,26)(27,33,35,31)(28,37,36,32)(34,39,40,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

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, 8679852);