#8227323

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	#37912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8134519);