#8688350

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

5095 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,24,27)(19,22,23,25)(20,29,21,26)(28,32,31,30);(1,3,5,4)(2,6,7,8)(9,10,12,14)(11,13,16,15)(17,19,24,23)(18,25,27,22)(20,28,21,31)(26,32,29,30);(9,10)(11,13)(12,14)(15,16)(17,20,24,21)(18,26,27,29)(19,28,23,31)(22,30,25,32);(1,2)(3,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15)(17,21,24,20)(18,26,27,29)(19,28,23,31)(22,32,25,30);(1,2)(3,6)(4,8)(5,7)(17,22)(18,19)(20,30)(21,32)(23,27)(24,25)(26,28)(29,31);(1,4,5,3)(2,6,7,8)(17,23,24,19)(18,25,27,22)(20,31,21,28)(26,32,29,30)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8595546);