#6487255

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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

1233 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,32,36)(27,39,44,53)(28,45,46,51)(29,35,40,50)(30,38,31,37)(33,43,52,42)(34,41,54,47)(48,56,49,55);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,43,32,44,30,42)(26,33,38,53,36,52,37,39)(28,29,47,49,46,40,41,48)(34,35,45,56,54,50,51,55);(11,17)(14,21)(18,19)(22,23)(25,28,31,47,32,46,30,41)(26,34,38,45,36,54,37,51)(27,40,43,48,44,29,42,49)(33,50,53,55,52,35,39,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,32,40)(26,35,36,50)(27,41,44,47)(28,42,46,43)(30,48,31,49)(33,51,52,45)(34,39,54,53)(37,55,38,56);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(26,36)(28,46)(29,40)(33,52)(37,38)(39,53)(41,47)(48,49)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394451);