#8734794

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

4870 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=g,e^2=h,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, 8641990);