#6377427

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34643	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1537 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)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,26,22,25,32,38)(19,23,29,35,28,34,41,45)(20,24,31,37,30,36,42,46)(27,33,39,44,40,43,47,48);(1,3,6,2)(4,7,5,8)(17,19,21,29,22,28,32,41)(18,23,26,35,25,34,38,45)(20,27,31,39,30,40,42,47)(24,33,37,44,36,43,46,48);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,36)(26,37)(28,40)(29,39)(32,42)(34,43)(35,44)(38,46)(41,47)(45,48);(1,5,6,4)(2,7,3,8)(19,28)(20,30)(23,34)(24,36)(29,41)(31,42)(35,45)(37,46);(9,11,12,15)(10,14,13,16)(18,25)(23,34)(24,36)(26,38)(33,43)(35,45)(37,46)(44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=f,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,d^b=di,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6284623);