#8199665

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

5162 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:

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

Polycyclic presentation:

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

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, 8106861);