#8534621

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	#40780	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4255 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=h,e^2=i,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, 8441817);