#6718318

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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)(3,7)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(17,18,22,27)(19,24,31,42)(20,33,36,25)(21,26,39,23)(28,45,47,41)(29,30,38,32)(34,46,48,43)(35,40,44,37);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,24)(20,34)(21,38)(22,31)(23,30)(25,43)(26,32)(27,42)(28,37)(29,39)(33,46)(35,45)(36,48)(40,47)(41,44);(10,13)(14,16)(17,20,22,36)(18,25,27,33)(19,28,31,47)(21,35,39,44)(23,40,26,37)(24,41,42,45)(29,46,38,43)(30,34,32,48);(2,6)(5,7)(10,13)(14,16)(17,21)(18,26)(19,29)(20,35)(22,39)(23,27)(24,30)(25,37)(28,46)(31,38)(32,42)(33,40)(34,41)(36,44)(43,47)(45,48);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(19,30)(21,39)(23,26)(24,38)(28,34)(29,42)(31,32)(35,44)(37,40)(41,43)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6625514);