#8424960

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

4769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,e^2=h,f^2=gh,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, 8332156);