#8757026

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

4432 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664222);