#8505576

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

4353 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8412772);