#6846513

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fhi,d^2=fg,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, 6753709);