#8459104

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	#40126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(20,28)(24,31)(27,32)(29,30)(33,34,39,42)(35,40,37,36)(38,41,47,46)(43,48,45,44);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,39,37)(34,36,42,40)(38,43,47,45)(41,44,46,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,34)(35,36)(37,40)(38,46)(39,42)(41,47)(43,48)(44,45);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,39,35)(34,40,42,36)(38,45,47,43)(41,48,46,44);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8366300);