#8607455

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	#51937	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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)(3,9)(4,13)(5,15)(6,8)(7,12)(10,11)(14,16)(17,18)(19,24)(20,21)(22,23)(25,30)(26,29)(27,31)(28,32)(34,39)(36,40)(41,47)(44,45);(1,3)(2,7)(4,14)(5,16)(6,12)(8,9)(10,15)(11,13)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,28)(31,32)(33,34,38,39)(35,40,43,36)(37,41,46,47)(42,45,48,44);(2,8)(5,13)(7,9)(11,16)(18,23)(20,30)(24,26)(27,32)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,45,47,44);(1,4)(2,5)(3,10)(6,15)(7,11)(8,13)(9,16)(12,14)(17,19)(18,24)(20,27)(21,31)(22,29)(23,26)(25,28)(30,32)(33,36)(34,35)(37,45)(38,40)(39,43)(41,48)(42,47)(44,46);(1,5)(2,4)(3,11)(6,13)(7,10)(8,15)(9,14)(12,16)(17,20)(18,25)(19,27)(21,23)(22,30)(24,28)(26,31)(29,32)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,47)(46,48);(1,3)(2,9)(4,14)(5,11)(6,12)(7,8)(10,15)(13,16)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,32,26,27)(33,37,38,46)(34,41,39,47)(35,42,43,48)(36,44,40,45)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8514651);