#8424946

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

4771 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332142);