#8757103

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

4345 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=gi,f^2=h,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, 8664299);