#6584541

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2033 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)(13,14,17,21)(15,19,23,26)(16,25,24,20)(18,22,27,28);(6,10)(11,12)(13,14)(15,20)(16,26)(17,21)(18,28)(19,24)(22,27)(23,25);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,15,17,23)(14,19,21,26)(16,27,24,18)(20,22,25,28);(5,8,9,7)(6,12,10,11)(15,16)(18,27)(19,25)(20,26)(22,28)(23,24);(13,16,17,24)(14,20,21,25)(15,18,23,27)(19,28,26,22)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491737);