#6408213

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	#34633	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1753 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6315409);