#6816689

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814799
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	#61287	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18)(19,23)(20,25)(21,24)(22,27)(26,33)(28,39)(29,37)(30,38)(31,36)(32,41)(34,40)(35,43)(42,46)(44,48)(45,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,23)(20,32)(21,28)(22,31)(24,39)(25,41)(26,38)(27,36)(29,34)(30,33)(35,44)(37,40)(42,47)(43,48)(45,46);(18,24,27,26)(19,28,31,30)(23,36)(25,40,43,42)(29,44,45,32)(37,47)(38,39)(41,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,20,33,46,22,35,21,34)(18,25,26,42,27,43,24,40)(19,29,28,44,31,45,30,32)(23,37,39,48,36,47,38,41)

Polycyclic presentation:

2^9|c^2=ghi,d^2=f,e^2=fhi,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

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, 6723885);