#8926845

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

4427 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)(11,15)(12,17)(13,21)(14,16)(18,22)(19,20)(23,24)(25,26)(27,31)(28,34)(29,33)(30,32)(35,37)(36,40)(38,39);(3,7)(6,8)(9,11)(10,15)(12,16)(13,19)(14,17)(18,23)(20,21)(22,24)(26,27)(29,37)(32,34)(36,38);(1,2)(3,6)(4,5)(7,8)(10,16)(11,17)(13,22)(20,23)(25,27,31,26)(28,34,30,32)(29,35,37,33)(36,40,38,39);(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,23)(22,24)(25,28,31,30)(26,32,27,34)(29,36,37,38)(33,39,35,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,16)(13,20)(14,15)(18,24)(19,21)(22,23)(25,29,31,37)(26,33,27,35)(28,36,30,38)(32,39,34,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)(25,30,31,28)(26,34,27,32)(29,38,37,36)(33,40,35,39)

Polycyclic presentation:

2^9|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,d^c=dgi,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, 8834041);