#9237006

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

5303 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)(11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,23)(24,27)(26,29)(28,30)(31,32);(1,2)(3,7)(4,5)(6,8)(9,11)(10,16)(12,14)(13,15)(18,23)(21,24)(25,30)(29,31);(2,5)(6,7)(9,10,15,14)(11,16,13,12)(20,28)(25,30)(26,32)(29,31);(9,12)(10,13)(11,14)(15,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,32)(29,30);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,16,14,12)(17,20,22,28)(18,25,23,30)(19,26,27,32)(21,29,24,31);(1,3)(2,6)(4,8)(5,7)(9,14,15,10)(11,16,13,12)(17,21,22,24)(18,19,23,27)(20,29,28,31)(25,26,30,32)

Polycyclic presentation:

2^9|c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300624
maximal normal subgroup of	#72300624
maximal quotient of	#72300637, #72300944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9144202);