#7197546

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1475 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)(17,18,22,28,23,27,38,45)(19,29,32,40,31,46,37,24)(20,25,35,42,34,43,48,39)(21,26,30,44,36,41,33,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19,23,31)(18,24,27,40)(20,33,34,30)(21,35,36,48)(22,37,38,32)(25,41,43,26)(28,46,45,29)(39,47,42,44);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,20,23,34)(18,25,27,43)(19,30,31,33)(21,32,36,37)(22,35,38,48)(24,26,40,41)(28,42,45,39)(29,44,46,47);(3,8)(5,7)(10,11)(14,15)(17,21,22,30,23,36,38,33)(18,26,28,44,27,41,45,47)(19,20,32,35,31,34,37,48)(24,39,29,25,40,42,46,43);(9,12,13,16)(10,14,11,15)(18,27)(19,31)(20,34)(26,41)(28,45)(32,37)(35,48)(44,47)

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,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7104742);