#8924761

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,18,22,26)(19,27,29,23)(20,24,28,30)(21,25,32,31);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,19,22,29)(18,23,26,27)(20,21,28,32)(24,31,30,25)(33,34,39,42)(35,43,37,40)(36,38,41,48)(44,47,46,45);(1,2,5,8)(3,6,11,9)(4,12,13,7)(10,15,16,14)(17,18,22,26)(19,23,29,27)(20,24,28,30)(21,31,32,25)(33,35,39,37)(34,40,42,43)(36,46,41,44)(38,47,48,45);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,28)(27,29)(31,32)(33,36,39,41)(34,38,42,48)(35,44,37,46)(40,45,43,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,20,22,28)(18,24,26,30)(19,21,29,32)(23,31,27,25)(33,37,39,35)(34,40,42,43)(36,44,41,46)(38,47,48,45);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21)(18,25)(19,28)(20,29)(22,32)(23,24)(26,31)(27,30)(33,38)(34,41)(35,45)(36,42)(37,47)(39,48)(40,46)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831957);