#242277

order	512
nilpotency class	3
isoclinism	isoclinic class of #233958
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

744 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,21)(14,17)(16,18)(20,23)(22,24)(25,26,29,34)(27,35,37,31)(28,33,30,32)(36,40,38,39);(2,5)(6,8)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19)(25,27,29,37)(26,31,34,35)(28,38,30,36)(32,39,33,40);(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19)(26,32)(27,36)(28,30)(33,34)(37,38)(39,40);(1,3,4,7)(2,6,5,8)(25,28,29,30)(26,33,34,32)(27,36,37,38)(31,39,35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,d^c=di,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, 149473);