#6718308

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,18)(19,28)(20,32)(21,25)(22,27)(23,30)(24,33)(26,36)(29,43)(31,41)(34,39)(35,38)(37,48)(40,47)(42,44)(45,46);(10,13)(14,16)(17,19,22,31)(18,23,27,38)(20,29,33,44)(21,35,36,30)(24,37,32,46)(25,28,26,41)(34,48,47,45)(39,43,40,42);(2,6)(5,7)(10,13)(14,16)(17,20)(18,24)(19,29)(21,34)(22,33)(23,37)(25,40)(26,39)(27,32)(28,42)(30,45)(31,44)(35,48)(36,47)(38,46)(41,43);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(20,33)(21,36)(23,28)(24,39)(26,27)(29,44)(30,35)(32,40)(37,43)(38,41)(42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=i,e^2=g,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, 6625504);