#8709231

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

5096 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,e^2=g,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, 8616427);