#8633366

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	#34681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4816 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8540562);