#6410815

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	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=fg,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cg,d^b=dfi,d^c=di,e^c=ei,e^d=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, 6318011);