#7514255

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421451);