#8137613

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

4107 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044809);