#6670190

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	#35761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=i,d^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, 6577386);