#6411122

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

1223 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)(7,13)(11,16)(12,15)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,31)(29,32)(33,34,39,43,38,41,37,42)(35,40,44,47,36,45,46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31)(33,35)(34,40)(36,38)(37,46)(39,44)(41,45)(42,48)(43,47);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)(34,41)(40,45)(42,43)(47,48);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(18,24)(19,27)(23,28)(26,29)(33,36,38,35)(34,40,41,45)(37,44,39,46)(42,48,43,47);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,24)(22,25)(23,28)(31,32)(33,37,38,39)(34,42,41,43)(35,44,36,46)(40,47,45,48)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi,e^b=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, 6318318);