#6409245

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

1805 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)(11,15)(12,14)(13,16)(17,18,22,26,21,25,34,40)(19,27,30,42,29,23,35,37)(20,31,33,45,32,24,38,39)(28,36,44,47,43,41,46,48);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,41)(32,43)(33,44)(34,35)(38,46)(39,47)(40,42)(45,48);(10,14)(11,16)(17,20)(18,24)(19,28)(21,32)(22,33)(23,36)(25,31)(26,39)(27,41)(29,43)(30,44)(34,38)(35,46)(37,47)(40,45)(42,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(18,25)(19,29)(20,32)(26,40)(30,35)(33,38)(36,41)(47,48);(2,5)(6,8)(10,14)(13,15)(18,25)(19,29)(24,31)(26,40)(28,43)(30,35)(39,45)(44,46)

Polycyclic presentation:

2^9|a^2=h,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,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, 6316441);