#8533594

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	#56877	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,4)(5,12)(6,9)(7,8)(10,11)(13,14)(15,16)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,7,5)(2,4,8,12)(6,10,15,13)(9,11,16,14)(20,28)(25,30)(27,29)(31,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,8)(3,5)(9,16)(10,13)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4)(2,3)(5,8)(6,14)(7,12)(9,13)(10,16)(11,15)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,5,7,3)(2,4,8,12)(6,13,15,10)(9,11,16,14)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31);(1,6,7,15)(2,9,8,16)(3,10,5,13)(4,11,12,14)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440790);