#242683

order	512
nilpotency class	3
isoclinism	isoclinic class of #242681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,g^b=gh,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149879);