#8924882

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

4407 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

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, 8832078);