#250186

order	512
nilpotency class	3
isoclinism	isoclinic class of #250071
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	#18888	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1014 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157382);