#6443193

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 6350389);