#6686765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#42890	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1945 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)(5,6,9,11)(7,12,8,10)(13,14)(15,20)(16,21)(17,24)(18,25)(19,23)(22,27)(26,28);(5,6)(7,10)(8,12)(9,11)(13,15,19,24)(14,17,23,20)(16,21,26,28)(18,25,27,22);(1,2,3,4)(13,16)(14,21)(15,18)(17,22)(19,26)(20,25)(23,28)(24,27);(5,7,9,8)(6,10,11,12)(13,17,19,20)(14,15,23,24)(16,25,26,22)(18,21,27,28);(5,8,9,7)(6,10,11,12)(13,18)(14,22)(15,16)(17,21)(19,27)(20,28)(23,25)(24,26)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=g,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593961);