#6537059

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535696
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×AGL(1,ℤ/8)	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1539 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6444255);