#6438827

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,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, 6346023);