#6445313

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2058 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,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, 6352509);