#6534892

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#537	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2636 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,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, 6442088);