#6495967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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

1508 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18)(19,27)(20,25)(21,33)(22,24)(23,29)(26,34)(28,40)(30,39)(31,37)(32,45)(35,43)(36,42)(38,44)(41,47)(46,48);(9,10)(11,14)(12,13)(15,16)(17,19,21,29,22,30,34,43)(18,23,26,27,24,35,33,39)(20,28,32,42,31,41,44,48)(25,36,38,40,37,46,45,47);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(18,24)(20,31)(23,35)(26,33)(27,39)(28,41)(32,44)(42,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,22,31)(18,25,24,37)(19,28,30,41)(21,32,34,44)(23,36,35,46)(26,38,33,45)(27,40,39,47)(29,42,43,48);(9,11,12,15)(10,14,13,16)(20,31)(25,37)(28,41)(32,44)(36,46)(38,45)(40,47)(42,48)

Polycyclic presentation:

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

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, 6403163);