#6709438

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1260 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,29,33)(27,31,37,47)(28,39,41,32)(30,34,43,51)(35,53,54,45)(36,46,38,48)(40,56,55,49)(42,52,44,50);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,29,37)(26,31,33,47)(28,40,41,55)(30,36,43,38)(32,49,39,56)(34,46,51,48)(35,44,54,42)(45,52,53,50);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,35)(29,41)(30,44)(31,45)(33,39)(34,52)(36,40)(37,54)(38,55)(42,43)(46,49)(47,53)(48,56)(50,51);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(28,41)(32,39)(35,54)(40,55)(42,44)(45,53)(49,56)(50,52);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,36)(28,41)(30,43)(31,46)(32,39)(34,51)(35,55)(37,38)(40,54)(45,56)(47,48)(49,53)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=fgi,e^2=f,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,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, 6616634);