#237108

order	512
nilpotency class	3
isoclinism	isoclinic class of #237079
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

1271 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)(4,8,5,9)(7,13,16,12)(17,18,23,28)(19,29,33,40)(20,34,36,44)(21,27,22,26)(24,32,41,31)(25,37,43,38)(30,39,45,47)(35,48,46,42);(1,3)(2,7)(4,13)(5,12)(6,14)(8,11)(9,15)(10,16)(17,19,23,33)(18,24,28,41)(20,35,36,46)(21,32,22,31)(25,42,43,48)(26,40,27,29)(30,38,45,37)(34,47,44,39);(3,12)(4,5)(7,11)(8,9)(13,14)(15,16)(17,20)(18,25)(19,30)(21,37)(22,38)(23,36)(24,39)(26,44)(27,34)(28,43)(29,42)(31,35)(32,46)(33,45)(40,48)(41,47);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(20,36)(25,43)(30,45)(34,44)(35,46)(37,38)(39,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144304);