#250607

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
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
prime radical	#61220	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1067 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)(17,18,23,28)(19,24,31,40)(20,32,34,43)(21,27,22,26)(25,36,42,35)(29,41,45,48)(30,44,37,39)(33,38,46,47);(1,3,5,12)(2,6,9,15)(4,13,14,11)(7,10,8,16)(17,19)(18,24)(20,33)(21,30)(22,37)(23,31)(25,41)(26,39)(27,44)(28,40)(29,36)(32,47)(34,46)(35,45)(38,43)(42,48);(17,20,23,34)(18,25,28,42)(19,29,31,45)(21,36,22,35)(24,38,40,47)(26,43,27,32)(30,46,37,33)(39,48,44,41);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,19)(18,24)(20,29)(21,37)(22,30)(23,31)(25,38)(26,44)(27,39)(28,40)(32,48)(33,36)(34,45)(35,46)(41,43)(42,47)

Polycyclic presentation:

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

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, 157803);