#7563454

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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	C₂×#41	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1599 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)(9,10,15,19)(11,21,25,17)(12,26,30,28)(13,14,20,16)(18,29,36,31)(22,38,39,35)(23,24,32,33)(27,37,40,34);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,27)(13,32)(14,24)(15,25)(16,33)(18,35)(19,21)(20,23)(22,31)(26,34)(28,37)(29,39)(30,40)(36,38);(9,12,14,29,15,30,16,31)(10,18,13,28,19,36,20,26)(11,22,33,40,25,39,24,27)(17,34,23,38,21,37,32,35);(9,13,16,10,15,20,14,19)(11,23,24,17,25,32,33,21)(12,28,31,18,30,26,29,36)(22,38,27,34,39,35,40,37);(1,4,5,3)(2,7,8,6)(9,14,15,16)(10,13,19,20)(11,24,25,33)(12,29,30,31)(17,32,21,23)(18,28,36,26)(22,27,39,40)(34,35,37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=g,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,g^b=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470650);