#6437534

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6130 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)(11,21,24,31)(12,26,29,33)(13,19,14,18)(16,23,32,25)(17,28,34,27)(22,36,38,30)(35,37,40,39);(2,4)(5,6)(7,8)(9,11)(10,16)(12,22)(13,23)(14,25)(15,24)(17,30)(18,31)(19,21)(20,32)(26,40)(27,37)(28,39)(29,38)(33,35)(34,36);(6,8)(9,12,15,29)(10,17,20,34)(11,22,24,38)(13,27,14,28)(16,30,32,36)(18,26,19,33)(21,35,31,40)(23,37,25,39);(6,8)(9,13,15,14)(10,18,20,19)(11,23,24,25)(12,27,29,28)(16,31,32,21)(17,26,34,33)(22,37,38,39)(30,40,36,35);(11,24)(16,32)(21,31)(22,38)(23,25)(30,36)(35,40)(37,39)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315569, #72315572, #72315591
maximal normal subgroup of	#72315569, #72315572, #72315591
maximal quotient of	#72315795

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344730);