#6846495

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fh,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, 6753691);