#8690952

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

5162 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8598148);