#8159299

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

4673 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,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, 8066495);