#7563204

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

2750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=gi,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^c=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, 7470400);