#8828872

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

4985 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8736068);