#6462678

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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

1371 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)(17,18,22,28,23,27,34,39)(19,29,31,26,32,44,21,24)(20,25,37,42,36,43,45,47)(30,40,33,41,46,48,35,38);(2,5)(6,8)(17,19,23,32)(18,24,27,26)(20,33,36,35)(21,34,31,22)(25,40,43,48)(28,44,39,29)(30,45,46,37)(38,47,41,42);(10,13)(14,16)(17,20,34,45,23,36,22,37)(18,25,39,47,27,43,28,42)(19,30,21,35,32,46,31,33)(24,38,44,48,26,41,29,40);(1,3,4,7)(2,6,5,8)(17,21,22,19,23,31,34,32)(18,26,28,44,27,24,39,29)(20,35,37,30,36,33,45,46)(25,41,42,48,43,38,47,40);(9,11,12,15)(10,14,13,16)(18,27)(20,36)(24,26)(28,39)(29,44)(30,46)(33,35)(37,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369874);