#6431315

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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, prime radical	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,22)(20,27)(21,24)(23,29)(25,32)(26,31)(28,30)(33,34,37,42,38,40,46,48)(35,39,36,44,43,47,45,41);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)(33,35,37,36,38,43,46,45)(34,39,42,44,40,47,48,41);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(18,24)(22,31)(23,27)(30,32)(34,40)(39,47)(41,44)(42,48);(2,10)(7,11)(8,9)(15,16)(33,36,37,43,38,45,46,35)(34,41,42,39,40,44,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338511);