#8395030

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

4512 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)(3,6)(4,5)(7,8)(25,26,31,32)(27,34,28,33)(29,37,39,35)(30,36,40,38);(1,2,4,5)(3,6,7,8)(9,10,14,15)(11,17,20,16)(12,21,23,13)(18,24,22,19)(26,32)(29,39)(33,34)(36,38);(3,7)(6,8)(10,15)(13,21)(16,17)(19,24)(25,27,31,28)(26,33,32,34)(29,36,39,38)(30,35,40,37);(2,5)(6,8)(9,11,14,20)(10,16,15,17)(12,22,23,18)(13,19,21,24)(25,28,31,27)(26,34,32,33)(29,36,39,38)(30,35,40,37);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,18)(14,23)(15,21)(16,24)(17,19)(20,22)(25,29,31,39)(26,35,32,37)(27,36,28,38)(30,33,40,34);(1,3,4,7)(2,6,5,8)(9,13)(10,12)(11,19)(14,21)(15,23)(16,22)(17,18)(20,24)(25,30)(26,36)(27,37)(28,35)(29,33)(31,40)(32,38)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=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, 8302226);