#8472869

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

4344 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=gh,f^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, 8380065);