#8119547

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=g,e^2=gi,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, 8026743);