#8197939

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

4847 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8105135);