#7470831

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1488 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7378027);