#6630885

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629011
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

1606 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)(9,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,24)(21,35)(22,27)(23,33)(25,34)(26,29)(30,38)(31,45)(32,39)(36,40)(37,46)(41,43)(42,47)(44,48);(2,5)(6,8)(17,19,29,30,22,32,20,33)(18,23,24,39,27,38,26,28)(21,31,43,44,36,46,34,47)(25,37,40,48,41,45,35,42);(10,13)(14,16)(17,20,22,29)(18,24,27,26)(19,30,32,33)(21,34,36,43)(23,28,38,39)(25,40,41,35)(31,44,46,47)(37,42,45,48);(1,3,4,7)(2,6,5,8)(17,21,20,34,22,36,29,43)(18,25,26,35,27,41,24,40)(19,31,33,47,32,46,30,44)(23,37,28,42,38,45,39,48);(9,11,12,15)(10,14,13,16)(19,32)(23,38)(28,39)(30,33)(31,46)(37,45)(42,48)(44,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ci,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538081);