#9040968

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	#35348	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5224 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,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, 8948164);