#8159006

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	#55086	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3996 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 8066202);