#237114

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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1121 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)(3,10,11,15)(4,12,13,7)(5,14,16,8)(17,18,23,28)(19,29,33,40)(20,34,38,43)(21,26,22,27)(24,31,41,32)(25,37,44,36)(30,39,46,47)(35,48,45,42);(1,3,6,11)(2,4,9,13)(5,12,16,7)(8,15,14,10)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,42)(26,29)(27,40)(28,41)(30,37)(34,47)(36,46)(38,45)(39,43)(44,48);(1,4)(2,7)(3,8)(5,15)(6,13)(9,12)(10,16)(11,14)(17,20)(18,25)(19,30)(21,37)(22,36)(23,38)(24,39)(26,34)(27,43)(28,44)(29,42)(31,45)(32,35)(33,46)(40,48)(41,47);(3,11)(4,13)(7,12)(10,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,29,41,40)(25,43,44,34)(30,35,46,45)(39,48,47,42)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,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, 144310);