#6443253

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5007 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)(9,10,15,20)(11,21,24,17)(12,16,26,31)(13,18,27,33)(14,29,22,19)(23,35,37,32)(25,30,38,40)(28,39,36,34);(1,2,3,4)(5,6)(7,8)(9,11,22,26,15,24,14,12)(10,16,29,17,20,31,19,21)(13,23,36,38,27,37,28,25)(18,30,39,32,33,40,34,35);(6,8)(9,12,14,24,15,26,22,11)(10,17,19,16,20,21,29,31)(13,25,28,37,27,38,36,23)(18,32,34,30,33,35,39,40);(6,8)(9,13)(10,18)(11,23)(12,25)(14,28)(15,27)(16,30)(17,32)(19,34)(20,33)(21,35)(22,36)(24,37)(26,38)(29,39)(31,40);(13,27)(18,33)(23,37)(25,38)(28,36)(30,40)(32,35)(34,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=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, 6350449);