#8550971

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

4803 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8458167);