#6819978

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1132 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)(3,6)(4,8)(5,7)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,29,33)(27,35,38,31)(28,32,42,48)(30,34,37,44)(36,49,50,43)(39,47,41,45)(40,46,52,55)(51,56,53,54);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(27,36)(28,40)(30,37)(31,43)(32,46)(34,44)(35,49)(38,50)(39,41)(42,52)(45,47)(48,55);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,37,50,29,38,30,36)(26,31,44,49,33,35,34,43)(28,41,52,51,42,39,40,53)(32,47,55,54,48,45,46,56);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(27,38)(31,35)(36,50)(39,41)(43,49)(45,47)(51,53)(54,56);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,40,29,42,37,52)(26,32,34,46,33,48,44,55)(27,39,36,51,38,41,50,53)(31,45,43,54,35,47,49,56)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fgh,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6727174);