#10422184

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	#56516	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)(3,7,12,5)(4,8,14,16)(10,15,11,13)(17,18,23,21)(19,24,30,29)(20,26,31,25)(22,32,28,27)(33,34,39,41)(35,37,43,36)(38,40,47,46)(42,45,48,44);(1,3,6,12)(2,7,9,5)(4,11,14,10)(8,13,16,15)(17,19,23,30)(18,24,21,29)(20,28,31,22)(25,32,26,27)(33,35,39,43)(34,36,41,37)(38,42,47,48)(40,44,46,45);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,20)(18,25)(19,28)(21,26)(22,30)(23,31)(24,32)(27,29)(33,36,39,37)(34,35,41,43)(38,44,47,45)(40,42,46,48);(1,4)(2,8)(3,11)(5,15)(6,14)(7,13)(9,16)(10,12)(17,20)(18,26)(19,28)(21,25)(22,30)(23,31)(24,27)(29,32)(33,34)(35,36)(37,43)(38,46)(39,41)(40,47)(42,45)(44,48);(1,3)(2,5)(4,10)(6,12)(7,9)(8,13)(11,14)(15,16)(17,21,23,18)(19,29,30,24)(20,26,31,25)(22,32,28,27)(33,37,39,36)(34,35,41,43)(38,45,47,44)(40,42,46,48);(1,5)(2,3)(4,13)(6,7)(8,10)(9,12)(11,16)(14,15)(17,22,23,28)(18,27,21,32)(19,20,30,31)(24,25,29,26)(33,38)(34,40)(35,42)(36,44)(37,45)(39,47)(41,46)(43,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10329380);