#8662868

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

4871 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8570064);