#6697391

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16);(10,13)(11,15)(17,18,22,27)(19,24,29,25)(20,30,31,32)(21,23,28,26);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,23)(20,31)(21,28)(24,30)(25,32)(26,27);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19,22,29)(18,24,27,25)(20,28,31,21)(23,30,26,32);(17,20,22,31)(18,25,27,24)(19,28,29,21)(23,30,26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=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, 6604587);