#8176304

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

5341 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)(4,5)(6,8)(17,18,23,25)(19,26,20,24)(21,31,32,22)(27,28,29,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,19,23,20)(18,24,25,26)(21,27,32,29)(22,28,31,30);(2,5)(6,7)(10,13)(14,15)(18,25)(22,31)(24,26)(28,30);(9,11)(10,14)(12,16)(13,15)(17,20,23,19)(18,26,25,24)(21,29,32,27)(22,30,31,28);(1,3)(2,6)(4,8)(5,7)(17,21)(18,22)(19,27)(20,29)(23,32)(24,28)(25,31)(26,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8083500);