#6571715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38026	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(22,26)(24,31)(27,33)(28,32)(30,35)(34,36);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,25,28)(23,27,29,34)(24,32,31,26)(30,36,35,33);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,29)(22,27,28,34)(24,30,31,35)(26,33,32,36);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(23,29)(27,34)(30,35)(33,36);(8,16)(11,18)(14,20)(17,19)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,f^b=fi,d^c=di,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, 6478911);