#8097793

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

4732 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004989);