#9078770

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

4929 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8985966);