#6846499

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1179 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)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,32)(28,33)(29,34)(30,44)(31,36)(35,38)(37,45)(39,46)(40,47)(41,48)(42,49)(43,51)(50,53)(52,55)(54,56);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,31,39)(26,32,36,46)(28,40,43,53)(29,38,41,30)(33,47,51,50)(34,35,48,44)(37,52,54,42)(45,55,56,49);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,41,52,31,43,29,42)(26,33,48,55,36,51,34,49)(27,37,38,53,39,54,30,40)(32,45,35,50,46,56,44,47);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29,31,41)(26,34,36,48)(27,38,39,30)(28,42,43,52)(32,35,46,44)(33,49,51,55)(37,53,54,40)(45,50,56,47);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,30,29,39,31,38,41,27)(26,35,34,32,36,44,48,46)(28,40,42,54,43,53,52,37)(33,50,49,45,51,47,55,56)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fhi,d^2=fgi,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6753695);