#8256617

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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:

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

Polycyclic presentation:

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

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, 8163813);