#6630884

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

1614 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)(19,29)(20,25)(21,35)(22,27)(23,28)(24,33)(26,36)(30,39)(31,38)(32,40)(34,42)(37,41)(43,44)(45,48)(46,47);(2,5)(6,8)(17,19,22,30,23,32,20,33)(18,24,25,40,28,39,27,29)(21,31,36,44,37,45,34,46)(26,38,35,47,42,48,41,43);(10,13)(14,16)(17,20,23,22)(18,25,28,27)(19,30,32,33)(21,34,37,36)(24,29,39,40)(26,35,42,41)(31,44,45,46)(38,43,48,47);(1,3,4,7)(2,6,5,8)(17,21,20,34,23,37,22,36)(18,26,27,41,28,42,25,35)(19,31,33,46,32,45,30,44)(24,38,29,43,39,48,40,47);(9,11,12,15)(10,14,13,16)(19,32)(24,39)(29,40)(30,33)(31,45)(38,48)(43,47)(44,46)

Polycyclic presentation:

2^9|b^2=h,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, 6538080);