#7045046

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×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1302 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fg,d^2=fi,e^2=i,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, 6952242);