#293642

order	512
nilpotency class	3
isoclinism	isoclinic class of #271515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

817 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)(9,10,13,16)(11,17,19,24)(12,21,18,15)(14,22,23,20)(25,26)(27,34)(28,38)(29,33)(30,32)(31,37)(35,40)(36,39);(2,5)(6,7)(25,27,29,37)(26,31,33,34)(28,36,30,35)(32,39,38,40);(9,11,18,22,13,19,12,20)(10,14,21,24,16,23,15,17)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(1,3,4,8)(2,6,5,7)(11,19)(14,23)(17,24)(20,22)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gi,d^c=dhi,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, 200838);