#6442915

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#35150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2655 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=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, 6350111);