#8196393

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103589);