#6741839

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 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)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,22,27)(19,29,32,24)(20,34,36,25)(21,26,23,28)(30,40,33,38)(31,39,45,43)(35,42,37,41)(44,47,46,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,27,29)(20,31,36,45)(21,33,23,30)(25,39,34,43)(26,40,28,38)(35,44,37,46)(41,47,42,48);(10,14)(13,15)(19,30)(21,23)(24,38)(26,28)(29,40)(31,44)(32,33)(35,37)(39,47)(41,42)(43,48)(45,46);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,20)(18,25)(19,31)(21,37)(22,36)(23,35)(24,39)(26,42)(27,34)(28,41)(29,43)(30,44)(32,45)(33,46)(38,47)(40,48);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(17,21)(18,26)(19,32)(20,35)(22,23)(24,29)(25,41)(27,28)(34,42)(36,37)(44,46)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6649035);