#8424569

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

4936 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=gh,e^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, 8331765);