#317240

order	512
nilpotency class	3
isoclinism	isoclinic class of #317219
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

509 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)(3,7)(4,9)(5,8)(6,10)(11,16)(12,15)(13,14)(17,18,24,28)(19,29,33,38)(20,34,37,39)(21,23,27,22)(25,32,42,31)(26,40,44,36)(30,41,47,48)(35,45,46,43);(1,3)(2,7)(4,13)(5,12)(6,11)(8,15)(9,14)(10,16)(17,19,22,31,24,33,23,32)(18,25,27,29,28,42,21,38)(20,35,40,30,37,46,36,47)(26,43,39,41,44,45,34,48);(2,8)(3,11)(4,5)(7,14)(9,10)(15,16)(17,20)(18,26)(19,30)(21,39)(22,36)(23,40)(24,37)(25,41)(27,34)(28,44)(29,43)(31,35)(32,46)(33,47)(38,45)(42,48);(1,4,6,5)(2,9,10,8)(3,12,11,13)(7,15,16,14)(17,21,22,18,24,27,23,28)(19,29,31,42,33,38,32,25)(20,34,36,44,37,39,40,26)(30,45,35,41,47,43,46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=ehi,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,d^b=di,f^b=fh,d^c=di,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224436);