#6846284

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1320 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)(3,7)(4,8)(5,6)(10,13)(11,15)(17,18,21,25)(19,27,30,23)(20,32,35,24)(22,26,29,37)(28,41,42,36)(31,38,34,40)(33,45,44,39)(43,47,46,48);(9,10)(11,14)(12,13)(15,16)(19,28)(20,33)(22,29)(23,36)(24,39)(26,37)(27,41)(30,42)(31,34)(32,45)(35,44)(38,40);(2,6)(7,8)(10,13)(14,16)(17,19,29,42,21,30,22,28)(18,23,37,41,25,27,26,36)(20,34,44,43,35,31,33,46)(24,40,45,47,32,38,39,48);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(19,30)(23,27)(28,42)(31,34)(36,41)(38,40)(43,46)(47,48);(1,4,5,3)(2,8,6,7)(17,20,22,33,21,35,29,44)(18,24,26,39,25,32,37,45)(19,31,28,43,30,34,42,46)(23,38,36,47,27,40,41,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,d^2=fgi,e^2=fghi,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,e^c=ei

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, 6753480);