#330765

order	512
nilpotency class	3
isoclinism	isoclinic class of #329395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

564 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=e,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237961);