#7514133

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

3144 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)(9,10,15,19)(11,20,13,27)(12,23,24,17)(14,29,30,18)(16,28,32,22)(21,35,25,34)(26,33,39,38)(31,36,37,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,19,32)(12,21,24,25)(14,28,30,22)(17,31,23,37)(18,27,29,20)(26,40,39,36)(33,35,38,34);(6,8)(9,12)(10,17)(11,21)(13,25)(14,26)(15,24)(16,31)(18,33)(19,23)(20,34)(22,36)(27,35)(28,40)(29,38)(30,39)(32,37);(2,4)(6,8)(12,24)(17,23)(21,25)(26,39)(31,37)(33,38)(34,35)(36,40);(9,13,15,11)(10,16,19,32)(12,25,24,21)(14,22,30,28)(17,31,23,37)(18,27,29,20)(26,36,39,40)(33,35,38,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,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, 7421329);