#7459768

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7366964);