#9027973

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

4829 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^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, 8935169);