#6585525

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
prime radical	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1856 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492721);