#6846450

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

1181 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)(25,26,31,37)(27,33,41,50)(28,42,45,34)(29,35,32,38)(30,36,40,49)(39,55,56,48)(43,54,47,51)(44,53,46,52);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,31,41)(26,33,37,50)(28,43,45,47)(29,40,32,30)(34,51,42,54)(35,49,38,36)(39,44,56,46)(48,52,55,53);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,29,44,31,45,32,46)(26,34,35,52,37,42,38,53)(27,39,30,47,41,56,40,43)(33,48,36,54,50,55,49,51);(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)(26,35,37,38)(27,40,41,30)(28,44,45,46)(33,49,50,36)(34,52,42,53)(39,43,56,47)(48,51,55,54);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,30,32,27,31,40,29,41)(26,36,38,33,37,49,35,50)(28,43,46,56,45,47,44,39)(34,51,53,55,42,54,52,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fhi,d^2=fg,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6753646);