#8712718

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

5141 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)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32);(2,7)(3,4)(11,15)(14,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32);(1,3,6,4)(2,8,7,5)(9,10,13,12)(11,14,15,16)(20,29)(24,28)(27,30)(31,32);(1,4,6,3)(2,5,7,8)(9,11,13,15)(10,14,12,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32);(1,5)(2,3)(4,7)(6,8)(9,12,13,10)(11,16,15,14)(17,21,22,19)(18,23,25,26)(20,30,29,27)(24,31,28,32);(17,18)(19,23)(20,28)(21,26)(22,25)(24,29)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8619914);