#8108304

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

4574 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)(6,8)(10,15)(11,19)(12,21)(13,23)(25,26)(27,30)(28,32)(29,37)(31,34)(33,39)(35,40)(36,38);(2,5)(6,8)(9,10,14,15)(11,16,19,22)(12,20,21,17)(13,18,23,24)(25,27,31,28)(26,32,34,30)(29,35,39,36)(33,40,37,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,16,15,22)(12,18,21,24)(13,20,23,17)(25,28,31,27)(26,32,34,30)(29,36,39,35)(33,40,37,38);(9,12,14,21)(10,17,15,20)(11,18,19,24)(13,22,23,16)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,22)(20,21)(23,24)(25,30,31,32)(26,27,34,28)(29,38,39,40)(33,35,37,36);(1,3)(2,6)(4,7)(5,8)(9,13)(10,18)(11,20)(12,22)(14,23)(15,24)(16,21)(17,19)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015500);