#8900054

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

4290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8807250);