#8161274

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	#52530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=gi,e^2=gh,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, 8068470);