#8274282

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

4585 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8181478);