#302258

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

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, 209454);