#8724824

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

4885 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8632020);