#6578847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2085 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)(14,19)(20,28)(21,23)(22,27);(5,7,9,8)(6,10,12,11)(13,14,17,19)(15,23,25,21)(16,20,24,28)(18,27,26,22);(2,4)(5,7,9,8)(6,11,12,10)(13,15,17,25)(14,20,19,28)(16,18,24,26)(21,27,23,22);(5,7,9,8)(6,11,12,10)(15,24)(16,25)(18,26)(20,21)(22,27)(23,28);(5,8,9,7)(6,10,12,11)(13,16)(14,21)(15,18)(17,24)(19,23)(20,27)(22,28)(25,26)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,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	#72250887, #72257154, #72257188, …
maximal normal subgroup of	#72250887, #72257154, #72257188, …
maximal quotient of	#72264785

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486043);