#7514918

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23912	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7422114);