#6694301

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1156 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6601497);