#9062556

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4609 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8969752);