#6459264

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,26,22,27,30,23)(19,28,29,35,32,24,37,39)(20,25,33,40,34,41,44,36)(31,42,43,46,45,38,47,48);(2,6)(7,8)(18,23)(19,29)(20,33)(21,30)(26,27)(31,45)(32,37)(34,44)(35,39)(36,40)(38,48)(42,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,30,37,22,32,21,29)(18,24,23,35,27,28,26,39)(20,31,44,47,34,45,33,43)(25,38,36,46,41,42,40,48);(1,4,5,3)(2,8,6,7)(17,20,21,33,22,34,30,44)(18,25,26,40,27,41,23,36)(19,31,29,43,32,45,37,47)(24,38,39,48,28,42,35,46);(9,11,12,15)(10,14,13,16)(19,32)(24,28)(29,37)(31,45)(35,39)(38,42)(43,47)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366460);