#6495246

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1613 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18)(19,27)(20,31)(21,35)(22,25)(23,30)(24,33)(26,37)(28,44)(29,42)(32,43)(34,41)(36,39)(38,47)(40,45)(46,48);(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,23,25,39)(20,32,33,28)(21,36,37,30)(24,40,31,38)(26,42,35,27)(34,45,48,47)(41,43,46,44);(2,5)(6,8)(10,13)(14,16)(17,20,22,33)(18,24,25,31)(19,28,29,32)(21,34,37,48)(23,38,39,40)(26,41,35,46)(27,43,42,44)(30,45,36,47);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(20,33)(23,39)(24,31)(27,42)(30,36)(34,48)(41,46);(9,11,12,15)(10,14,13,16)(18,25)(23,39)(24,31)(26,35)(27,42)(38,40)(41,46)(43,44)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di

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, 6402442);