#6411008

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	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1169 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,31)(29,32)(33,34,38,42,37,39,36,41)(35,43,45,48,44,40,46,47);(1,3,6,12)(2,7,9,10)(4,11,5,13)(8,15,14,16)(17,19,22,30)(18,23,26,27)(20,29,21,28)(24,32,25,31)(35,44)(40,43)(45,46)(47,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,27,32)(34,39)(35,44)(41,42)(45,46);(3,12)(7,10)(11,13)(15,16)(17,21,22,20)(18,25,26,24)(19,29,30,28)(23,32,27,31)(33,35,37,44)(34,40,39,43)(36,46,38,45)(41,48,42,47);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,26)(23,27)(24,25)(31,32)(33,36,37,38)(34,41,39,42)(35,45,44,46)(40,47,43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=g,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi,d^c=di,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, 6318204);