#243904

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

655 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:

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

Polycyclic presentation:

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

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, 151100);