#8285307

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	#49414	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8192503);