#6836172

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1329 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,36)(27,32,39,48)(28,33,37,46)(29,34,40,49)(30,43,45,35)(38,53,54,47)(41,52,44,50)(42,56,55,51);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,32)(28,40)(29,37)(30,41)(31,39)(33,49)(34,46)(35,50)(36,48)(38,42)(43,52)(44,45)(47,51)(53,56)(54,55);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,40,27,31,37,29,39)(26,33,49,32,36,46,34,48)(30,44,55,54,45,41,42,38)(35,52,56,53,43,50,51,47);(10,16)(14,17)(15,22)(21,23)(25,29,31,40)(26,34,36,49)(27,37,39,28)(30,42,45,55)(32,46,48,33)(35,51,43,56)(38,44,54,41)(47,52,53,50);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,30,40,55,31,45,29,42)(26,35,49,56,36,43,34,51)(27,38,37,44,39,54,28,41)(32,47,46,52,48,53,33,50)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fghi,d^2=i,e^2=ghi,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, 6743368);