#6462647

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1377 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)(17,18,22,27,23,28,35,40)(19,29,31,26,32,45,21,24)(20,33,38,48,37,25,41,44)(30,46,34,39,47,42,36,43);(2,5)(6,8)(17,19,23,32)(18,24,28,26)(20,34,37,36)(21,35,31,22)(25,42,33,46)(27,45,40,29)(30,41,47,38)(39,48,43,44);(10,13)(14,16)(17,20,35,41,23,37,22,38)(18,25,40,48,28,33,27,44)(19,30,21,36,32,47,31,34)(24,39,45,46,26,43,29,42);(1,3,4,7)(2,6,5,8)(17,21,22,19,23,31,35,32)(18,26,27,45,28,24,40,29)(20,36,38,30,37,34,41,47)(25,43,44,46,33,39,48,42);(9,11,12,15)(10,14,13,16)(20,37)(25,33)(30,47)(34,36)(38,41)(39,43)(42,46)(44,48)

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,d^a=di,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, 6369843);