#6457840

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365036);