#279402

order	512
nilpotency class	3
isoclinism	isoclinic class of #267131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

611 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 186598);