#8618043

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8525239);