#6557899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=f,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=ei,e^d=eh

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, 6465095);