#6481964

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389160);