#8712377

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	#40886	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8619573);