#6584837

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	#36471	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1969 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)(13,14)(15,20)(16,21)(17,19)(18,22)(23,27)(24,28)(25,26);(6,10)(11,12)(14,19)(15,16)(18,24)(20,27)(21,26)(23,25);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,28)(22,27);(5,8,9,7)(6,12,10,11)(15,23)(16,25)(18,24)(20,27)(21,26)(22,28);(13,16,17,23)(14,21,19,27)(15,24,25,18)(20,28,26,22)

Polycyclic presentation:

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492033);