#8919239

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

4819 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)(11,13)(12,15)(14,16)(17,18)(19,24)(20,30)(21,25)(22,26)(23,29)(27,28)(31,32);(3,7)(6,8)(9,10,14,16)(11,13,15,12)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,31)(30,32);(1,2,5,4)(3,6,7,8)(9,11,14,15)(10,12,16,13)(20,25)(21,30)(27,31)(28,32);(9,12,14,13)(10,11,16,15)(17,20,22,25)(18,21,26,30)(19,27,29,31)(23,28,24,32);(1,3,5,7)(2,6,4,8)(9,10)(11,12)(13,15)(14,16)(17,19)(18,24)(20,31)(21,28)(22,29)(23,26)(25,27)(30,32);(1,4,5,2)(3,8,7,6)(9,13,14,12)(10,15,16,11)(17,21)(18,25)(19,28)(20,26)(22,30)(23,31)(24,27)(29,32)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826435);