#8650964

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

4822 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=i,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8558160);