#8119335

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

5007 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8026531);