#6846803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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)(27,37)(28,33)(29,44)(30,34)(31,36)(32,39)(35,40)(38,53)(41,52)(42,49)(43,50)(45,51)(46,48)(47,55)(54,56);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,30,39)(26,32,34,37)(28,41,42,46)(29,45,40,31)(33,48,49,52)(35,51,44,36)(38,54,55,43)(47,56,53,50);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,31,43,30,42,45,54)(26,33,36,50,34,49,51,56)(27,38,40,41,39,55,29,46)(32,47,44,48,37,53,35,52);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(26,34)(28,42)(32,37)(35,44)(36,51)(38,55)(41,46)(43,54);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,45,39,30,40,31,27)(26,35,51,37,34,44,36,32)(28,41,54,38,42,46,43,55)(33,48,56,47,49,52,50,53)

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,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6753999);