#8517562

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	#41311	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8424758);