#8932876

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	#35390	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,e^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,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, 8840072);