#8159069

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

3848 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066265);