#8094982

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

4735 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,30)(28,33)(29,39)(31,34)(32,37)(35,40)(36,38);(2,5)(6,7)(11,17)(15,21)(18,23)(22,24)(25,27)(26,32)(28,35)(29,36)(30,34)(31,37)(33,38)(39,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,21)(16,19)(22,24)(25,28,31,29)(26,33,34,39)(27,35,37,36)(30,40,32,38);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,31,28)(26,33,34,39)(27,36,37,35)(30,40,32,38);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,30,31,32)(26,27,34,37)(28,38,29,40)(33,36,39,35);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,27)(26,32)(28,35)(29,36)(30,34)(31,37)(33,38)(39,40)

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002178);