#6514427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6421623);