#8159223

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	#41260	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,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, 8066419);