#8829018

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

5181 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:

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

Polycyclic presentation:

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

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, 8736214);