#6836163

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

1327 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)(27,38)(28,42)(29,37)(30,46)(31,36)(32,40)(33,41)(34,43)(35,48)(39,49)(44,50)(45,51)(47,52)(53,55)(54,56);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,32)(28,29)(30,44)(31,40)(33,37)(34,42)(35,50)(36,38)(39,55)(41,43)(45,56)(46,52)(47,48)(49,51)(53,54);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,43,40,31,41,29,27)(26,33,34,38,36,42,37,32)(30,47,55,39,48,44,45,56)(35,52,51,49,46,50,53,54);(10,16)(14,17)(15,22)(21,23)(25,29,31,43)(26,34,36,37)(27,28,40,41)(30,45,48,55)(32,42,38,33)(35,51,46,53)(39,44,56,47)(49,52,54,50);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,30,43,55,31,48,29,45)(26,35,34,51,36,46,37,53)(27,39,28,44,40,56,41,47)(32,49,33,50,38,54,42,52)

Polycyclic presentation:

2^9|b^2=fi,c^2=fghi,d^2=i,e^2=gh,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, 6743359);