#8284175

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

5076 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)(4,6)(5,8)(11,15)(13,16)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28);(1,3)(2,5)(4,8)(6,7)(9,10,12,14)(11,13,15,16)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32);(2,6)(5,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,25)(24,32)(26,27);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,26)(21,30)(22,29)(23,32)(25,27)(28,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30);(1,2)(3,5)(4,6)(7,8)(17,19)(18,21)(20,26)(22,25)(23,31)(24,30)(27,29)(28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^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, 8191371);