#8174459

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5078 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,15,13,16)(17,18)(19,21)(20,25)(22,30)(23,24)(26,31)(27,32)(28,29);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(17,19)(18,21)(20,23)(22,27)(24,25)(26,29)(28,31)(30,32);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,16,12,15)(18,24)(19,20)(26,30)(27,28);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,32,30,29);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,15,12,16)(17,21,23,25)(18,20,24,19)(22,29,31,32)(26,28,30,27);(17,22,23,31)(18,26,24,30)(19,27,20,28)(21,29,25,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081655);