#6499078

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1270 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,19)(13,15)(16,20)(17,21)(18,24)(22,23)(25,26,30,34)(27,36,29,32)(28,39,42,55)(31,45,40,35)(33,48,49,43)(37,52,41,54)(38,47,44,51)(46,53,50,56);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,22)(19,24)(20,23)(25,27,30,29)(26,32,34,36)(28,37,42,41)(31,38,40,44)(33,46,49,50)(35,47,45,51)(39,54,55,52)(43,53,48,56);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,19)(18,23)(25,28,40,48,30,42,31,43)(26,33,45,55,34,49,35,39)(27,37,44,56,29,41,38,53)(32,46,51,52,36,50,47,54);(1,4,5,3)(2,8,6,7)(11,17)(14,21)(18,23)(22,24)(25,29,30,27)(26,32,34,36)(28,41,42,37)(31,44,40,38)(33,46,49,50)(35,47,45,51)(39,54,55,52)(43,56,48,53);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(27,29)(32,36)(37,41)(38,44)(46,50)(47,51)(52,54)(53,56)

Polycyclic presentation:

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

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, 6406274);