#8734954

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

4997 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)(10,12)(14,15)(17,18)(19,26)(20,22)(21,24)(23,25)(27,28)(29,31)(30,32);(1,2,5,8)(3,6,4,7)(9,10,13,12)(11,15,16,14)(17,19,23,21)(18,24,25,26)(20,29,30,27)(22,28,32,31);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,12,15)(18,25)(20,30)(24,26)(27,29);(1,2)(3,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,23,30)(18,22,25,32)(19,27,21,29)(24,31,26,28);(1,4,5,3)(2,7,8,6)(17,21,23,19)(18,26,25,24)(20,29,30,27)(22,28,32,31);(17,22)(18,20)(19,28)(21,31)(23,32)(24,29)(25,30)(26,27)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,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, 8642150);