#9040361

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

4959 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)(3,4)(5,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32);(1,3,5,7)(2,6,8,4)(9,10,14,16)(11,13,15,12)(18,19)(20,27)(24,26)(28,32);(1,2,5,8)(3,6,7,4)(9,11,14,15)(10,12,16,13)(20,27)(21,29)(23,25)(24,26);(1,2,5,8)(3,4,7,6)(9,12,14,13)(10,11,16,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,2)(3,6)(4,7)(5,8)(9,10)(11,12)(13,15)(14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,4,5,6)(2,7,8,3)(9,13,14,12)(10,15,16,11)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^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=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947557);