#7186451

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1239 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)(12,20)(15,21)(19,24)(22,23)(25,26,29,31)(27,35,37,38)(28,33,32,40)(30,36,39,34);(2,5)(3,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(26,31)(28,32)(34,36)(35,38);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,27,30,33,29,37,39,40)(26,32,34,35,31,28,36,38);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(25,28)(26,33)(27,36)(29,32)(30,38)(31,40)(34,37)(35,39);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(27,37)(28,32)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=fi,e^2=h,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=dh,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, 7093647);