#8436223

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,e^2=g,f^2=g,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, 8343419);