#6443554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2304 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)(13,14,19,23)(15,20,27,38)(16,21,30,42)(17,31,18,22)(24,34,25,35)(26,37,32,36)(28,39,43,44)(29,41,33,40);(6,9)(10,12)(13,15,19,27)(14,20,23,38)(16,28,30,43)(17,26,18,32)(21,39,42,44)(22,36,31,37)(24,33,25,29)(34,40,35,41);(1,2,3,4)(13,16)(14,21)(15,24)(17,29)(18,33)(19,30)(20,34)(22,40)(23,42)(25,27)(26,43)(28,32)(31,41)(35,38)(36,39)(37,44);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,29)(18,33)(19,30)(20,35)(22,40)(23,42)(24,27)(26,28)(31,41)(32,43)(34,38)(36,44)(37,39);(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,29,30,33)(20,36,38,37)(21,40,42,41)(24,43,25,28)(34,39,35,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=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, 6350750);