#6462727

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,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,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369923);