#6412076

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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)(15,20)(19,24)(21,23)(25,26,32,35,31,36,30,38)(27,37,29,34,39,33,40,28);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,15,17,18)(12,19,13,21)(16,23,22,24)(25,27,31,39)(26,33,36,37)(28,38,34,35)(29,30,40,32);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,30,33,31,34,32,37)(26,27,38,40,36,39,35,29);(2,6)(5,8)(11,20)(15,18)(19,21)(23,24)(25,29)(26,34)(27,30)(28,36)(31,40)(32,39)(33,35)(37,38);(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(25,30,31,32)(26,35,36,38)(27,29,39,40)(28,33,34,37)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6319272);