#8434451

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=h,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, 8341647);