#8908391

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	#47996	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8815587);