#10040816

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

4526 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9948012);