#6693513

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1321 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6600709);