#8097224

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

4603 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)(9,10)(11,18)(12,20)(13,16)(14,17)(15,22)(19,21)(23,24)(25,26,31,30)(27,32,37,36)(28,38,29,33)(34,40,35,39);(1,2,4,6)(3,8,7,5)(9,11,14,16)(10,13,17,18)(12,19,22,23)(15,21,20,24)(25,27,31,37)(26,32,30,36)(28,34,29,35)(33,39,38,40);(1,3,4,7)(2,5,6,8)(25,28,31,29)(26,33,30,38)(27,34,37,35)(32,39,36,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,19)(13,21)(14,22)(16,23)(17,20)(18,24)(25,27)(26,32)(28,34)(29,35)(30,36)(31,37)(33,39)(38,40);(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,22)(23,24)(25,29,31,28)(26,33,30,38)(27,35,37,34)(32,39,36,40);(9,13)(10,16)(11,17)(12,21)(14,18)(15,23)(19,20)(22,24)(25,30,31,26)(27,36,37,32)(28,38,29,33)(34,40,35,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004420);