#232861

order	512
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#60907	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi

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, 140057);