#237165

order	512
nilpotency class	3
isoclinism	isoclinic class of #237155
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

1157 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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=dhi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144361);