#6575567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2134 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)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(13,14,18,22)(15,23,26,40)(16,27,30,38)(17,31,33,21)(19,32,36,24)(20,29,39,34)(25,37,28,35)(41,43,44,42);(2,4)(13,15,18,26)(14,19,22,36)(16,28,30,25)(17,32,33,24)(20,37,39,35)(21,40,31,23)(27,41,38,44)(29,43,34,42);(5,7,8,11)(6,10,9,12)(15,24)(16,29)(17,33)(19,23)(20,38)(21,31)(26,32)(27,39)(30,34)(36,40)(41,44)(42,43);(13,16,18,30)(14,20,22,39)(15,25,26,28)(17,34,33,29)(19,35,36,37)(21,27,31,38)(23,41,40,44)(24,42,32,43)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482763);