#320739

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

630 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,27,29,32,38,34)(28,35,37,40,36,33,39,31);(2,5)(6,7)(9,11,13,20,12,19,21,18)(10,14,16,23,15,17,24,22)(25,27,30,32,29,34,38,26)(28,31,37,35,36,40,39,33);(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,28,29,36)(26,31,32,40)(27,33,34,35)(30,39,38,37);(1,3,4,8)(2,6,5,7)(10,15)(14,17)(16,24)(22,23)(26,32)(27,34)(31,40)(33,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,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, 227935);