#6660447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660434
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35251	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi

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, 6567643);