#6432276

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2266 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)(13,14)(15,19)(16,20)(17,22)(18,21)(23,32)(24,34)(25,33)(26,35)(27,37)(28,36)(29,39)(30,40)(31,38)(41,43)(42,44);(6,9)(10,12)(13,15,18,25)(14,19,21,33)(16,26,28,42)(17,29,31,24)(20,35,36,44)(22,39,38,34)(23,30,41,27)(32,40,43,37);(2,4)(13,16,18,28)(14,20,21,36)(15,23,25,41)(17,30,31,27)(19,32,33,43)(22,40,38,37)(24,42,29,26)(34,44,39,35);(5,7,8,11)(6,10,9,12);(14,21)(19,33)(20,36)(22,38)(32,43)(34,39)(35,44)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339472);