#237324

order	512
nilpotency class	3
isoclinism	isoclinic class of #237305
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1128 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:

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

Polycyclic presentation:

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

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, 144520);