#8137616

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	#46088	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 8044812);