#8285419

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	#35782	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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)(19,21)(23,26)(27,30)(31,32)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,47,45);(1,3,7,11)(2,8,10,6)(4,12,9,16)(5,13,15,14)(17,18)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,45)(44,48)(46,47);(3,11)(6,8)(12,16)(13,14)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)(33,36,38,39)(34,35,41,43)(37,45,46,44)(40,47,48,42);(1,4)(2,5)(3,12)(6,14)(7,9)(8,13)(10,15)(11,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,47)(46,48);(1,5)(2,9)(3,13)(4,10)(6,12)(7,15)(8,16)(11,14)(17,21,22,19)(18,23,25,26)(20,27,29,30)(24,32,28,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,45)(44,48)(46,47);(1,6)(2,3)(4,14)(5,12)(7,8)(9,13)(10,11)(15,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,29)(30,31)(33,37,38,46)(34,40,41,48)(35,42,43,47)(36,44,39,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,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, 8192615);