#6506434

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2333 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)(17,18)(19,21)(20,27)(22,25)(23,26)(24,29)(28,31)(30,32);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,26,23,25,21,18)(20,28,29,24,31,27,30,32);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,30,23,31,22,29)(18,24,25,28,26,32,19,27);(11,16)(13,15)(17,21,23,22)(18,19,26,25)(20,29,31,30)(24,28,32,27);(10,14)(13,15)(17,21,23,22)(18,19,26,25)(20,30,31,29)(24,27,32,28)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh

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, 6413630);