#8934729

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	#35419	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=h,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, 8841925);