#6605894

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513090);