#6477258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384454);