#6553840

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^d=ehi

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, 6461036);