#8497908

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

4588 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)(11,16)(12,17)(13,18)(14,19)(15,20)(21,24)(22,23)(25,26,30,31)(27,32,29,34)(28,36,37,33)(35,40,38,39);(1,2,4,6)(3,5,7,8)(11,20)(12,22)(15,16)(17,23)(26,31)(32,34)(33,36)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,21)(13,17)(14,20)(16,19)(18,22)(23,24)(25,27,30,29)(26,32,31,34)(28,35,37,38)(33,39,36,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,21)(13,23)(14,20)(15,19)(17,24)(18,22)(25,28)(26,33)(27,35)(29,38)(30,37)(31,36)(32,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,21)(13,15)(14,22)(16,24)(18,20)(19,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(9,13,14,24)(10,18,19,21)(11,17,20,23)(12,15,22,16)(25,29,30,27)(26,34,31,32)(28,35,37,38)(33,39,36,40)

Polycyclic presentation:

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

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, 8405104);