#9175011

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

5186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9082207);