#8391337

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

4781 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)(3,4)(5,6)(7,8)(17,18,23,25)(19,26,20,22)(21,29,31,24)(27,30,28,32);(2,6)(3,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,25)(19,26,20,22)(21,29,31,24)(27,30,28,32);(2,6)(4,7)(10,13)(14,15)(17,19,23,20)(18,22,25,26)(21,27,31,28)(24,30,29,32);(2,6)(3,8)(9,11,12,16)(10,14,13,15)(17,20,23,19)(18,22,25,26)(21,27,31,28)(24,32,29,30);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,21,23,31)(18,24,25,29)(19,27,20,28)(22,30,26,32);(1,4,5,7)(2,3,6,8)(9,10)(11,14)(12,13)(15,16)(17,22)(18,19)(20,25)(21,30)(23,26)(24,27)(28,29)(31,32)

Polycyclic presentation:

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

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, 8298533);