#8285487

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	#52415	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(17,18)(19,29)(20,26)(21,22)(23,27)(24,28)(25,32)(30,31)(33,34)(35,44)(36,38)(37,46)(39,43)(40,47)(41,48)(42,45);(1,3,7,11)(2,8,10,6)(4,12,9,16)(5,13,15,14)(17,19)(18,25)(20,30)(21,27)(22,31)(23,26)(24,32)(28,29)(33,35)(34,40)(36,45)(37,48)(38,46)(39,47)(41,42)(43,44);(3,11)(6,8)(12,16)(13,14)(17,20,24,21)(18,26,28,22)(19,30,32,27)(23,29,31,25)(33,36,39,48)(34,41,43,38)(35,45,47,37)(40,42,44,46);(1,4)(2,5)(3,12)(6,14)(7,9)(8,13)(10,15)(11,16)(17,21,24,20)(18,26,28,22)(19,27,32,30)(23,29,31,25)(33,37)(34,42)(35,36)(38,40)(39,45)(41,44)(43,46)(47,48);(1,5)(2,9)(3,13)(4,10)(6,12)(7,15)(8,16)(11,14)(17,22,24,26)(18,21,28,20)(19,31,32,23)(25,27,29,30)(33,35)(34,40)(36,45)(37,48)(38,46)(39,47)(41,42)(43,44);(1,6)(2,3)(4,14)(5,12)(7,8)(9,13)(10,11)(15,16)(17,23,24,31)(18,27,28,30)(19,26,32,22)(20,25,21,29)(33,38,39,41)(34,36,43,48)(35,46,47,42)(37,40,45,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gh,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, 8192683);