#6448129

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2011 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:

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

Polycyclic presentation:

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

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, 6355325);