#8701749

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

5590 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608945);