#6453096

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6360292);