#6576489

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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

2752 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)(7,12)(8,10)(9,11)(13,14,17,19)(15,20,24,25)(16,21,27,23)(18,22,28,26);(5,7,9,8)(6,10,11,12)(14,19)(15,23)(16,25)(18,28)(20,27)(21,24);(2,4)(13,15)(14,20)(16,26)(17,24)(18,21)(19,25)(22,27)(23,28);(5,8,9,7)(6,10,11,12)(15,21)(16,25)(18,28)(20,27)(22,26)(23,24);(13,16,17,27)(14,21,19,23)(15,22,24,26)(18,20,28,25)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483685);