#7198262

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=fghi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105458);