#302802

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=f,d^2=ei,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=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209998);