#317181

order	512
nilpotency class	3
isoclinism	isoclinic class of #317171
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

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

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,d^c=di,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, 224377);