#250641

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	#18218	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1070 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^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, 157837);