#8824459

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

4607 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731655);