#8968528

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

4789 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8875724);