#6670834

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	#32913	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1903 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)(5,6,8,9)(7,11,12,10)(13,14,18,17)(15,19,23,22)(16,20,25,26)(21,27,28,24);(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,19,17,22)(16,24,25,27)(20,21,26,28);(2,4)(6,9)(10,11)(13,16,18,25)(14,20,17,26)(15,21,23,28)(19,27,22,24);(5,7,8,12)(6,10,9,11)(16,25)(20,26)(21,28)(24,27);(14,17)(15,22)(19,23)(20,26)(21,24)(27,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=f,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, 6578030);