#6561896

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6469092);