#242735

order	512
nilpotency class	3
isoclinism	isoclinic class of #242735
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

741 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)(12,17,28,33,25,32,26,34)(21,35,38,30,24,39,37,40);(9,11,13,22,14,23,15,29)(10,16,18,20,19,31,27,36)(12,24,26,38,25,21,28,37)(17,30,33,39,32,40,34,35);(2,5)(3,8)(6,7)(9,12)(10,17)(11,21)(13,28)(14,25)(15,26)(16,30)(18,34)(19,32)(20,35)(22,38)(23,24)(27,33)(29,37)(31,40)(36,39);(2,6)(5,7)(12,25)(17,32)(21,24)(26,28)(30,40)(33,34)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,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,d^c=dh,e^c=eh,f^c=fi

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, 149931);