#6514426

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	#37553	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1833 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)(15,25,27,37)(16,30,32,40)(17,23,18,22)(20,29,36,28)(21,33,39,34)(26,35,31,38)(41,42,44,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,29,19,27,17,28)(14,20,23,37,24,36,22,25)(16,31,34,42,32,26,33,43)(21,38,30,41,39,35,40,44);(2,4)(6,9)(7,11)(13,16,17,33,19,32,18,34)(14,21,22,40,24,39,23,30)(15,26,28,42,27,31,29,43)(20,35,25,41,36,38,37,44);(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)(15,27)(16,32)(20,36)(21,39)(25,37)(28,29)(30,40)(33,34)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,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, 6421622);