#8550989

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	#38645	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=h,e^2=h,f^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, 8458185);