#8436786

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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)(19,27)(20,24)(21,25)(22,26)(23,28)(29,32)(30,31)(33,34)(35,41)(36,43)(37,40)(38,39)(42,45)(44,48)(46,47);(1,2,6,4)(3,7,9,10)(5,8,14,12)(11,15,13,16)(19,28)(20,30)(23,27)(24,31)(34,39)(36,41)(40,47)(44,45);(17,19,22,28)(18,23,26,27)(20,29,30,21)(24,25,31,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,3)(2,7)(4,10)(5,13)(6,9)(8,16)(11,14)(12,15)(17,20)(18,24)(19,29)(21,28)(22,30)(23,25)(26,31)(27,32)(33,36)(34,35)(37,45)(38,41)(39,43)(40,48)(42,47)(44,46);(1,4,6,2)(3,10,9,7)(5,12,14,8)(11,16,13,15)(17,21,22,29)(18,25,26,32)(19,20,28,30)(23,31,27,24);(1,5,6,14)(2,8,4,12)(3,11,9,13)(7,15,10,16)(17,18)(19,23)(20,31)(21,25)(22,26)(24,30)(27,28)(29,32)(33,37,38,46)(34,40,39,47)(35,42,43,48)(36,44,41,45)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,e^2=gh,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, 8343982);