#8121438

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	#56809	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8028634);