#242921

order	512
nilpotency class	3
isoclinism	isoclinic class of #238015
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 150117);