#8157729

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

4949 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8064925);