#6767616

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23965	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 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)(4,6)(5,8)(17,18,22,27)(19,29,31,24)(20,26,21,35)(23,38,37,28)(25,36,43,34)(30,40,33,44)(32,41,47,45)(39,46,42,48);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,32)(25,42)(26,40)(27,29)(28,41)(34,48)(35,44)(36,46)(37,47)(38,45)(39,43);(10,13)(14,16)(17,20,22,21)(18,25,27,43)(19,30,31,33)(23,36,37,34)(24,39,29,42)(26,28,35,38)(32,48,47,46)(40,45,44,41);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,20)(18,26,27,35)(19,30,31,33)(23,36,37,34)(24,40,29,44)(25,28,43,38)(32,46,47,48)(39,45,42,41);(3,8)(5,7)(10,13)(14,16)(20,21)(25,43)(26,35)(30,33)(34,36)(39,42)(40,44)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6674812);