#237523

order	512
nilpotency class	3
isoclinism	isoclinic class of #237505
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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1199 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=di,e^c=ei,f^c=fhi,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, 144719);