#305002

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212198);