#235186

order	512
nilpotency class	3
isoclinism	isoclinic class of #235159
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1248 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)(3,11,14,15)(4,7,5,9)(8,12,16,13)(17,18,23,28)(19,29,33,39)(20,34,36,42)(21,26,22,27)(24,32,40,31)(25,37,43,35)(30,41,46,38)(44,47,48,45);(2,7)(4,5)(8,15)(9,10)(11,16)(12,13)(17,19,23,33)(18,24,28,40)(20,30,36,46)(21,32,22,31)(25,41,43,38)(26,29,27,39)(34,44,42,48)(35,47,37,45);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,15,9,11)(17,20,23,36)(18,25,28,43)(19,30,33,46)(21,35,22,37)(24,38,40,41)(26,42,27,34)(29,44,39,48)(31,45,32,47);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,42,43,34)(30,45,46,47)(38,48,41,44)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=hi,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,e^c=ei,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142382);