#8496892

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	#37332	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404088);