#6431767

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
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×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7198 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)(11,21,23,32)(12,26,29,33)(13,19,14,18)(16,24,31,25)(17,28,34,27)(22,36,37,30)(35,38,40,39);(2,4)(5,6)(7,8)(9,11)(10,16)(12,22)(13,24)(14,25)(15,23)(17,30)(18,32)(19,21)(20,31)(26,40)(27,38)(28,39)(29,37)(33,35)(34,36);(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,26)(19,33)(20,34)(21,35)(23,37)(24,38)(25,39)(31,36)(32,40);(11,23)(16,31)(21,32)(22,37)(24,25)(30,36)(35,40)(38,39);(9,13,15,14)(10,18,20,19)(11,24,23,25)(12,27,29,28)(16,32,31,21)(17,26,34,33)(22,38,37,39)(30,40,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314709, #72314786, #72315547, …
maximal normal subgroup of	#72314709, #72314786, #72315547, …
maximal quotient of	#72316005, #72316033

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338963);