#8806911

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

4937 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)(3,5)(4,8)(6,7)(9,10,14,12)(11,16,15,13)(17,18)(19,22)(20,27)(21,23)(24,29)(25,31)(26,32)(28,30);(2,7)(3,8)(10,12)(13,16)(17,19,21,25)(18,22,23,31)(20,28,29,26)(24,32,27,30);(1,3,6,8)(2,4,7,5)(10,12)(11,15)(18,23)(19,25)(24,27)(26,28);(1,4,6,5)(2,3,7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,24)(21,23)(25,31)(26,30)(27,29)(28,32);(1,2)(3,4)(5,8)(6,7)(9,12,14,10)(11,16,15,13)(17,19)(18,22)(20,26)(21,25)(23,31)(24,30)(27,32)(28,29);(1,5,6,4)(2,8,7,3)(9,13,14,16)(10,15,12,11)(17,20,21,29)(18,24,23,27)(19,26,25,28)(22,30,31,32)

Polycyclic presentation:

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

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, 8714107);