#6744984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1194 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)(27,35)(28,40)(29,32)(30,34)(31,39)(33,41)(36,44)(37,45)(38,46)(42,53)(43,47)(48,54)(49,51)(50,56)(52,55);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,29,39)(26,31,32,35)(28,37,41,51)(30,36,43,38)(33,45,40,49)(34,46,47,44)(42,50,54,52)(48,56,53,55);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(27,36)(30,43)(31,44)(33,40)(35,46)(37,50)(38,39)(42,54)(45,55)(49,56)(51,52);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,40)(34,48)(35,49)(36,50)(38,52)(39,51)(43,54)(44,55)(46,56)(47,53);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,32)(27,38)(28,41)(30,43)(31,46)(35,44)(36,39)(37,50)(45,55)(48,53)(49,56)(51,52)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fi,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6652180);