#8924901

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

4387 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)(27,33)(28,34)(29,35)(30,36)(31,32)(37,38)(39,40);(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(26,32)(29,36)(33,34)(37,39);(1,3)(2,5)(4,7)(6,8)(10,14)(15,22)(16,19)(23,24)(25,27,31,28)(26,33,32,34)(29,37,36,39)(30,40,35,38);(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,28,31,27)(26,34,32,33)(29,37,36,39)(30,40,35,38);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,29,31,36)(26,35,32,30)(27,37,28,39)(33,38,34,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30,31,35)(26,36,32,29)(27,38,28,40)(33,37,34,39)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,e^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, 8832097);