#8734951

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

4831 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,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, 8642147);