#8773037

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	#58269	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8680233);