#317168

order	512
nilpotency class	3
isoclinism	isoclinic class of #317147
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

521 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)(3,10)(4,8)(5,7)(6,11)(9,13)(12,14)(15,16)(17,18,24,28)(19,29,34,31)(20,35,39,43)(21,23,27,22)(25,33,41,32)(26,37,44,38)(30,45,47,40)(36,42,48,46);(1,3)(2,6)(4,11)(5,12)(7,15)(8,10)(9,14)(13,16)(17,19,22,32,24,34,23,33)(18,25,27,29,28,41,21,31)(20,36,38,30,39,48,37,47)(26,42,35,40,44,46,43,45);(2,7)(3,11)(5,13)(6,14)(8,9)(10,15)(17,20)(18,26)(19,30)(21,35)(22,37)(23,38)(24,39)(25,40)(27,43)(28,44)(29,42)(31,46)(32,36)(33,48)(34,47)(41,45);(2,8)(6,10)(7,9)(14,15)(17,21,22,18,24,27,23,28)(19,31,32,25,34,29,33,41)(20,35,37,26,39,43,38,44)(30,46,36,40,47,42,48,45)

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,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, 224364);