#8862423

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	#39110	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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)(5,14)(6,8)(7,12)(10,16)(11,15)(20,27)(24,31)(26,28)(29,32)(34,36)(37,45)(39,41)(42,48);(1,3,6,12)(2,7,8,9)(4,10,14,15)(5,16,13,11)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,31)(28,32)(34,36)(37,45)(39,41)(42,48);(2,8)(4,14)(7,9)(10,15)(18,23)(19,25)(24,31)(26,28)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,4)(2,5)(3,10)(6,14)(7,16)(8,13)(9,11)(12,15)(17,19,22,25)(18,21,23,30)(20,28,27,26)(24,32,31,29)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,7)(4,15)(5,11)(6,12)(8,9)(10,14)(13,16)(17,20,22,27)(18,24,23,31)(19,26,25,28)(21,29,30,32)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,5)(2,4)(3,11)(6,13)(7,15)(8,14)(9,10)(12,16)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=i,e^2=i,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, 8769619);