#8285516

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	#41474	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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)(19,27)(20,30)(21,31)(22,26)(23,29)(24,28)(25,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,3,7,11)(2,8,10,6)(4,12,9,16)(5,13,15,14)(17,19,22,29)(18,23,26,27)(20,21,28,32)(24,31,30,25)(35,41)(38,45)(42,47)(46,48);(3,11)(6,8)(12,16)(13,14)(17,20,22,28)(18,24,26,30)(19,21,29,32)(23,31,27,25)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4)(2,5)(3,12)(6,14)(7,9)(8,13)(10,15)(11,16)(17,19)(18,23)(20,21)(22,29)(24,31)(25,30)(26,27)(28,32)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,5)(2,9)(3,13)(4,10)(6,12)(7,15)(8,16)(11,14)(17,21)(18,25)(19,28)(20,29)(22,32)(23,24)(26,31)(27,30)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,6)(2,3)(4,14)(5,12)(7,8)(9,13)(10,11)(15,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,28)(27,29)(31,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=gh,e^2=gh,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, 8192712);