#8119580

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	#40346	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026776);