#6554803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461999);