#242726

order	512
nilpotency class	3
isoclinism	isoclinic class of #242725
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

721 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149922);