#8459193

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

5348 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8366389);