#8517565

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	#58255	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,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, 8424761);