#9057806

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

5230 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)(4,8)(5,6)(10,12)(14,15)(17,18)(19,23)(20,27)(21,29)(22,24)(25,30)(26,32)(28,31);(2,6)(3,8)(9,10,13,12)(11,15,16,14)(17,19,22,20)(18,23,24,27)(21,26,30,28)(25,31,29,32);(1,3)(2,4)(5,8)(6,7)(9,11,13,16)(10,14,12,15)(18,24)(19,20)(21,30)(31,32);(1,4)(2,3)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,20,22,19)(18,23,24,27)(21,28,30,26)(25,31,29,32);(1,2)(3,4)(5,6)(7,8)(17,18)(19,23)(20,27)(21,25)(22,24)(26,31)(28,32)(29,30);(1,4)(2,3)(5,7)(6,8)(17,21,22,30)(18,25,24,29)(19,26,20,28)(23,31,27,32)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965002);