#242947

order	512
nilpotency class	3
isoclinism	isoclinic class of #238019
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

657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,8)(5,7)(9,13)(10,14)(11,15)(12,16)(17,18)(19,27,30,40)(20,24)(21,25)(22,26)(23,29,37,34)(28,43,31,42)(32,38)(33,39)(35,41)(36,45,47,44)(46,48);(3,9)(4,11)(6,13)(8,15)(10,12)(14,16)(17,19)(18,23)(20,31)(21,34)(22,30)(24,36)(25,40)(26,37)(27,41)(28,32)(29,35)(33,44)(38,47)(39,43)(42,48)(45,46);(1,3)(2,6)(4,12)(5,10)(7,14)(8,16)(9,11)(13,15)(17,20,21,33,22,32,35,46)(18,24,25,39,26,38,41,48)(19,28,29,44,30,31,34,45)(23,36,27,42,37,47,40,43);(2,7)(3,10)(8,15)(9,12)(20,32)(24,38)(28,31)(33,46)(36,47)(39,48)(42,43)(44,45)

Polycyclic presentation:

2^9|a^2=f,c^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=dhi,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, 150143);