#8750833

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

4873 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:

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

Polycyclic presentation:

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

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, 8658029);