#9124708

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
prime radical	#43014	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9031904);