#8436237

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	#59142	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343433);