#10106989

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

4738 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)(3,4)(5,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,23,27)(19,28,32,24)(20,21,25,31)(22,26,30,29);(1,3,5,7)(2,4,8,6)(9,10,13,12)(11,14,16,15)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,26)(27,28);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,20,23,25)(18,21,27,31)(19,29,32,26)(22,24,30,28);(1,3,5,7)(2,4,8,6)(9,11,13,16)(10,14,12,15)(17,18)(19,24)(20,21)(22,26)(23,27)(25,31)(28,32)(29,30);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,21)(18,25)(19,30)(20,27)(22,32)(23,31)(24,26)(28,29);(1,4)(2,7)(3,8)(5,6)(9,12,13,10)(11,15,16,14)(17,22,23,30)(18,26,27,29)(19,31,32,21)(20,24,25,28)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10014185);