#250489

order	512
nilpotency class	3
isoclinism	isoclinic class of #250455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1063 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)(3,10)(4,7)(5,9)(6,11)(8,14)(12,16)(13,15)(17,18,23,27)(19,28,31,38)(20,33,35,42)(21,36,22,26)(24,32,40,29)(25,37,41,34)(30,39,46,48)(43,44,47,45);(1,3)(2,6)(4,13)(5,12)(7,10)(8,16)(9,15)(11,14)(17,19)(18,24)(20,30)(21,32)(22,29)(23,31)(25,39)(26,38)(27,40)(28,36)(33,43)(34,44)(35,46)(37,45)(41,48)(42,47);(6,15)(10,16)(19,29)(21,22)(24,38)(26,36)(28,40)(30,44)(31,32)(33,42)(34,37)(39,47)(43,48)(45,46);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,20,23,35)(18,25,27,41)(19,30,31,46)(21,37,22,34)(24,39,40,48)(26,42,36,33)(28,43,38,47)(29,44,32,45)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157685);