#6846451

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

1177 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)(25,26,32,38)(27,33,41,49)(28,34,45,53)(29,35,31,37)(30,36,40,48)(39,47,55,56)(42,50,46,54)(43,51,44,52);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,32,41)(26,33,38,49)(28,42,45,46)(29,40,31,30)(34,50,53,54)(35,48,37,36)(39,43,55,44)(47,51,56,52);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,29,43,32,45,31,44)(26,34,35,51,38,53,37,52)(27,39,30,46,41,55,40,42)(33,47,36,54,49,56,48,50);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29,32,31)(26,35,38,37)(27,40,41,30)(28,43,45,44)(33,48,49,36)(34,51,53,52)(39,42,55,46)(47,50,56,54);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,30,31,27,32,40,29,41)(26,36,37,33,38,48,35,49)(28,42,44,55,45,46,43,39)(34,50,52,56,53,54,51,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fhi,d^2=fgi,e^2=fgh,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, 6753647);