#9012008

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

4804 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16);(3,4)(7,8)(9,10,12,14)(11,13,16,15)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(18,19)(20,27)(21,29)(30,31);(3,4)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,4,6,3)(2,7,5,8)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(1,5,6,2)(3,7,4,8)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|b^2=hi,d^2=ghi,e^2=gi,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^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, 8919204);