#6694033

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,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, 6601229);