#8137597

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

5256 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044793);