#8692583

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8599779);