#8271943

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

4925 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)(5,6)(7,8)(10,13)(11,12)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32);(2,6)(3,8)(9,10,14,13)(11,16,12,15)(19,20)(21,30)(23,26)(24,29);(2,6)(4,7)(9,11,14,12)(10,15,13,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,16,13,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32);(1,3)(2,7)(4,6)(5,8)(9,13,14,10)(11,16,12,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32);(1,4,5,7)(2,3,6,8)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=i,e^2=ghi,f^2=hi,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, 8179139);