#323584

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
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

622 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=egh,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230780);