#6578540

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=f,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,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	#72256952, #72256973, #72257011
maximal normal subgroup of	#72256952, #72256973, #72257011
maximal quotient of	#72264613, #72264762, #72264897, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485736);