#7604477

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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	#28243	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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:

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

Polycyclic presentation:

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

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, 7511673);