#317157

order	512
nilpotency class	3
isoclinism	isoclinic class of #317147
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

517 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,f^c=fi,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, 224353);