#8256581

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	#56909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18,22,23)(19,21,26,24)(20,28,29,25)(27,31,32,30)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,47,45,44);(2,7)(4,8)(9,14)(12,15)(18,23)(19,26)(20,29)(30,31)(33,35,38,43)(34,39,41,36)(37,42,46,45)(40,47,48,44);(5,13)(9,14)(10,16)(12,15)(18,23)(21,24)(25,28)(30,31)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,19,22,26)(18,24,23,21)(20,27,29,32)(25,31,28,30);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,21,22,24)(18,26,23,19)(20,30,29,31)(25,32,28,27)(33,37,38,46)(34,40,41,48)(35,42,43,45)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=h,e^2=ghi,f^2=hi,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, 8163777);