#6465873

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1869 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei

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, 6373069);