#6459272

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 6366468);