#6576539

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2911 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)(16,23)(20,24)(22,25)(26,27);(2,4)(13,14)(15,19)(16,24)(17,20)(18,23)(21,26)(22,28)(25,27);(5,8,9,7)(6,10,11,12)(14,18)(15,21)(16,23)(20,24)(22,25)(26,27);(13,15,17,22)(14,19,20,28)(16,25,23,21)(18,26,24,27)

Polycyclic presentation:

2^9|b^2=f,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,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281916, #72281943, #72282139
maximal normal subgroup of	#72281916, #72281943, #72282139

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483735);