#7197562

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1441 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=h,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

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, 7104758);