#8712709

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

4903 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)(9,10,13,11)(12,15,16,14)(17,18)(19,27)(20,30)(21,24)(22,28)(23,25)(26,31)(29,32);(3,7)(6,8)(10,11)(14,15)(17,19,23,21)(18,24,25,27)(20,28,31,29)(22,30,32,26);(1,2,5,4)(3,6,7,8)(9,11,13,10)(12,15,16,14)(18,25)(20,31)(24,27)(28,29);(1,3,5,7)(2,6,4,8)(9,12)(10,14)(11,15)(13,16)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,24,32,27);(1,4,5,2)(3,8,7,6)(17,21,23,19)(18,27,25,24)(20,28,31,29)(22,30,32,26);(17,22)(18,28)(19,30)(20,27)(21,26)(23,32)(24,31)(25,29)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8619905);