#6403185

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1334 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,31,36,30,35,27,32)(28,38,39,40,34,33,37,29);(1,3,4,7)(2,5,6,8)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(25,27,30,31)(26,32,35,36)(28,37,34,39)(29,33,40,38);(2,6)(5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28)(26,33)(27,37)(29,36)(30,34)(31,39)(32,40)(35,38);(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(28,34)(29,40)(33,38)(37,39);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,31,38,30,40,27,33)(26,34,36,37,35,28,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=df,d^c=di,e^c=ei,e^d=ei

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, 6310381);