#8161269

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

4988 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,15)(14,16)(18,22)(19,25)(24,30)(26,27);(1,3)(2,4)(5,8)(6,7)(10,11)(14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30);(1,3)(2,4)(5,8)(6,7)(9,10,13,11)(12,14,16,15)(19,25)(23,29)(26,27)(31,32);(2,6)(3,8)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,27)(21,25)(22,29)(24,32)(26,28)(30,31);(1,3)(2,4)(5,8)(6,7)(9,12)(10,14)(11,15)(13,16)(17,20,21,28)(18,24,22,30)(19,26,25,27)(23,31,29,32);(1,4)(2,3)(5,7)(6,8)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^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, 8068465);