#8658117

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

5135 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^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, 8565313);