#7563228

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2774 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:

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

Polycyclic presentation:

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

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, 7470424);