#8550924

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	#45508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=h,e^2=gh,f^2=gi,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, 8458120);