#6820550

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

1252 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fghi,d^2=f,e^2=hi,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, 6727746);