#295269

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,20,23)(12,21,16,13)(18,19,24,22)(25,26)(27,36)(28,40)(29,34)(30,44)(31,35)(32,38)(33,43)(37,54)(39,48)(41,47)(42,52)(45,49)(46,51)(50,55)(53,56);(2,5)(6,7)(9,11,14,20)(10,15,17,23)(12,18,16,24)(13,22,21,19)(25,27,29,38)(26,32,34,36)(28,41,42,56)(30,39,31,45)(33,50,51,54)(35,48,44,49)(37,46,55,43)(40,53,52,47);(9,12,13,10,14,16,21,17)(11,18,19,23,20,24,22,15)(25,28,30,43,29,42,31,46)(26,33,35,40,34,51,44,52)(27,37,39,56,38,55,45,41)(32,47,48,50,36,53,49,54);(1,3,4,8)(2,6,5,7)(26,34)(28,42)(32,36)(35,44)(37,55)(41,56)(43,46)(48,49)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202465);