#8276603

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
prime radical	#58991	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5236 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)(9,10,14,15)(11,13,12,16)(17,18)(19,24)(20,22)(21,25)(23,27)(26,31)(28,29)(30,32);(1,3)(2,5)(4,8)(6,7)(10,15)(11,12)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32);(1,2,4,6)(3,5,8,7)(9,11,14,12)(10,16,15,13)(21,29)(25,28)(26,32)(30,31);(1,3,4,8)(2,5,6,7)(9,12,14,11)(10,13,15,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,13)(10,11)(12,15)(14,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32);(1,2)(3,5)(4,6)(7,8)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)

Polycyclic presentation:

2^9|a^2=i,c^2=h,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, 8183799);