#7523117

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	#60593	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1700 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,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, 7430313);