#8425420

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332616);