#8154742

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

5400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,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, 8061938);