#6345079

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=di,e^a=eh,f^a=fh,f^b=fh,d^c=di,e^c=eh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252275);