#8724847

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

4970 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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 8632043);