#6555236

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1223 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6462432);