#8827959

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	#50658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(17,18,22,25)(19,23,26,20)(21,24,30,32)(27,31,29,28)(33,34,37,41)(35,38,43,39)(36,44,46,40)(42,48,45,47);(2,7)(4,13)(5,8)(11,14)(17,19,22,26)(18,23,25,20)(21,29,30,27)(24,28,32,31)(33,35)(34,38)(36,45)(37,43)(39,41)(40,48)(42,46)(44,47);(2,7)(3,10)(9,12)(11,14)(19,26)(20,23)(27,29)(28,31)(33,35,37,43)(34,39,41,38)(36,42,46,45)(40,48,44,47);(1,3,6,10)(2,8,7,5)(4,14,13,11)(9,15,12,16)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,47)(44,48)(45,46);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,41)(38,43)(42,48)(44,46)(45,47);(1,5,6,8)(2,10,7,3)(4,15,13,16)(9,14,12,11)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45)

Polycyclic presentation:

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

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, 8735155);