#6466815

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2178 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)(5,6,8,9)(7,11,12,10)(13,14)(15,23)(16,21)(17,19)(18,22)(20,27)(24,32)(25,31)(26,37)(28,36)(29,30)(33,41)(34,39)(35,40)(38,42)(43,44);(1,2,3,4)(6,9)(10,11)(14,19)(15,24)(16,28)(18,25)(20,32)(21,35)(23,34)(26,41)(27,39)(29,40)(30,36)(42,43);(5,7,8,12)(6,10,9,11)(13,15,25,39,17,27,18,24)(14,20,31,32,19,23,22,34)(16,26,40,44,29,41,28,38)(21,33,35,42,30,37,36,43);(13,16,25,40,17,29,18,28)(14,21,31,35,19,30,22,36)(15,26,39,44,27,41,24,38)(20,33,32,42,23,37,34,43);(15,27)(20,23)(24,39)(26,41)(32,34)(33,37)(38,44)(42,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gh,d^2=hi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374011);