#6576565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281916, #72281943, #72282179
maximal normal subgroup of	#72281916, #72281943, #72282179

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483761);