#8475382

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

4640 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,e^2=g,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, 8382578);