#9057771

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

4990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8964967);