#8734919

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	#54272	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,38,43)(35,41,44,39)(36,40,45,46)(37,42,47,48);(2,6)(7,9)(8,12)(15,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35)(34,39)(36,42)(37,40)(38,44)(41,43)(45,48)(46,47);(2,6)(7,9)(8,12)(15,16)(19,27)(20,28)(23,32)(24,29)(33,36,38,45)(34,40,43,46)(35,42,44,48)(37,41,47,39);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,36)(34,40)(35,42)(37,39)(38,45)(41,47)(43,46)(44,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,35)(34,41)(36,42)(37,46)(38,44)(39,43)(40,47)(45,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,37,38,47)(34,42,43,48)(35,40,44,46)(36,41,45,39)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8642115);