#6605912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,22,25,21)(20,24,28,31)(27,30,32,29);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,18,23,25,22,26)(20,27,29,24,28,32,30,31);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,19,22,26,23,25,21,18)(20,27,30,31,28,32,29,24);(9,11)(10,13)(12,15)(14,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,28)(25,32)(26,31);(11,15)(13,16)(20,28)(24,31)(27,32)(29,30)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fi,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=dhi,e^d=eh

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, 6513108);