#8750301

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

5084 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,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, 8657497);