#8288169

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

5199 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8195365);