#232875

order	512
nilpotency class	3
isoclinism	isoclinic class of #232862
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60961	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1143 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)(4,6)(5,15)(7,9)(8,12)(11,14)(13,16)(17,18)(19,29)(20,33)(21,28)(22,27)(23,26)(24,31)(25,35)(30,38)(32,43)(34,41)(36,42)(37,39)(40,46)(44,45)(47,48);(1,3)(2,8)(4,14)(5,11)(6,13)(7,12)(9,16)(10,15)(17,19)(18,24)(20,30)(21,37)(22,31)(23,32)(25,40)(26,39)(27,43)(28,29)(33,44)(34,47)(35,45)(36,46)(38,41)(42,48);(1,4,6,2)(3,10,13,16)(5,15,7,9)(8,12,14,11)(17,20,23,36)(18,25,26,41)(19,30,32,46)(21,35,22,34)(24,38,39,40)(27,33,28,42)(29,44,43,48)(31,47,37,45);(1,5,6,7)(2,9,4,15)(3,11,13,12)(8,16,14,10)(18,26)(24,39)(25,41)(27,28)(29,43)(33,42)(38,40)(44,48)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140071);