#8746604

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

4696 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)(3,8)(4,6)(5,7)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,25)(21,26)(22,27)(24,28)(29,31)(30,32);(1,3,6,5)(2,7,4,8)(10,11)(14,15)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,31)(26,32);(1,3,6,5)(2,7,4,8)(9,11,13,10)(12,15,16,14)(19,28)(23,24)(29,32)(30,31);(3,5)(7,8)(9,12,13,16)(10,14,11,15)(17,19)(18,24)(20,29)(21,31)(22,28)(23,27)(25,30)(26,32);(1,4,6,2)(3,8,5,7)(17,20)(18,25)(19,29)(21,27)(22,26)(23,31)(24,30)(28,32);(1,5,6,3)(2,7,4,8)(17,21,22,25)(18,26,27,20)(19,30,28,31)(23,32,24,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,e^2=h,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, 8653800);