#6684157

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6591353);