#8509541

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

5322 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416737);