#6449226

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,h^c=hi,f^d=fi,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, 6356422);