#8372277

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	#42606	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4172 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(33,34)(35,41)(36,46)(37,42)(38,40)(39,43)(44,48)(45,47);(2,7)(3,11)(4,14)(13,16)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31)(33,35,40,43)(34,41,38,39)(36,45,37,44)(42,48,46,47);(2,7)(5,9)(8,10)(13,16)(18,23)(21,25)(24,31)(27,32)(33,36,40,37)(34,42,38,46)(35,44,43,45)(39,48,41,47);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,32)(26,30)(33,37,40,36)(34,42,38,46)(35,45,43,44)(39,48,41,47);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32)(33,38,40,34)(35,41,43,39)(36,46,37,42)(44,47,45,48);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,21,22,25)(18,20,23,30)(19,27,28,32)(24,29,31,26)(33,39,40,41)(34,43,38,35)(36,47,37,48)(42,44,46,45)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8279473);