#8446086

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,e^2=ghi,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, 8353282);