#6674450

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1735 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)(3,11)(4,8)(5,10)(6,9)(7,13)(12,16)(14,15)(17,18)(19,30)(20,34)(21,39)(22,28)(23,29)(24,27)(25,33)(26,38)(31,46)(32,43)(35,48)(36,47)(37,44)(40,42)(41,45);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,25)(20,35)(21,32)(22,38)(23,33)(24,41)(26,46)(27,43)(28,47)(29,30)(31,36)(34,42)(37,40)(39,45)(44,48);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20)(18,26)(19,31)(21,36)(22,33)(23,37)(24,38)(25,42)(27,44)(28,45)(29,47)(30,48)(32,40)(34,39)(35,41)(43,46);(2,9)(7,11)(8,10)(15,16)(17,21,23,24)(18,27,29,39)(19,32,33,41)(20,36,37,38)(22,35,31,40)(25,43,30,45)(26,44,47,34)(28,42,46,48);(1,5,6,4)(2,10,9,8)(3,12,13,14)(7,15,11,16)(17,22,23,31)(18,28,29,46)(19,20,33,37)(21,40,24,35)(25,44,30,34)(26,45,47,43)(27,48,39,42)(32,38,41,36)

Polycyclic presentation:

2^9|c^2=fi,d^2=i,e^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,g^c=gi,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, 6581646);