#7563276

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

2752 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)(9,10,15,20)(11,21,24,18)(12,17,25,33)(13,16,29,32)(14,30,22,19)(23,34,38,36)(26,39,28,31)(27,40,37,35);(1,2)(3,4)(6,7)(9,11,22,29,15,24,14,13)(10,16,30,18,20,32,19,21)(12,23,37,26,25,38,27,28)(17,31,40,36,33,39,35,34);(5,6,8,7)(9,12)(10,17)(11,23)(13,28)(14,27)(15,25)(16,31)(18,36)(19,35)(20,33)(21,34)(22,37)(24,38)(26,29)(30,40)(32,39);(2,4)(5,7,8,6)(11,24)(12,25)(13,29)(16,32)(17,33)(18,21)(27,37)(35,40);(9,13,14,24,15,29,22,11)(10,18,19,16,20,21,30,32)(12,26,27,23,25,28,37,38)(17,34,35,39,33,36,40,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,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^c=ei,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, 7470472);