#6437568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

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

6104 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)(9,10,15,20)(11,21,25,29)(12,18,28,34)(13,16,14,19)(17,24,32,23)(22,35,38,39)(26,33,27,30)(31,36,40,37);(2,4)(5,6)(7,8)(10,16)(13,14)(17,29)(18,30)(19,20)(21,32)(23,24)(26,27)(31,39)(33,34)(35,40)(36,37);(6,8)(9,11,15,25)(10,17,20,32)(12,22,28,38)(13,23,14,24)(16,29,19,21)(18,31,34,40)(26,36,27,37)(30,39,33,35);(6,8)(9,12)(10,18)(11,22)(13,27)(14,26)(15,28)(16,30)(17,31)(19,33)(20,34)(21,35)(23,37)(24,36)(25,38)(29,39)(32,40);(9,13,15,14)(10,16,20,19)(11,23,25,24)(12,26,28,27)(17,29,32,21)(18,33,34,30)(22,36,38,37)(31,35,40,39)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=i,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^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 6344764);