#8167400

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

4861 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=i,e^2=gi,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, 8074596);