#9023645

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

4825 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,22,26)(20,21,25,31)(28,29,30,32);(3,8)(6,7)(9,10,12,14)(11,13,16,15)(17,19,23,22)(18,24,27,26)(20,28,25,30)(21,29,31,32);(1,3,5,8)(2,6,4,7)(9,10)(11,13)(12,14)(15,16)(19,22)(20,25)(21,31)(24,26);(3,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,21)(19,28)(22,30)(23,25)(24,29)(26,32)(27,31);(1,3)(2,7)(4,6)(5,8)(17,21)(18,25)(19,29)(20,27)(22,32)(23,31)(24,30)(26,28);(1,4,5,2)(3,6,8,7)(17,22,23,19)(18,26,27,24)(20,30,25,28)(21,32,31,29)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303333
maximal normal subgroup of	#72303333

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8930841);