#8932980

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
prime radical	#59184	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4822 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840176);