#6846382

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fh,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^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, 6753578);