#8617994

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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)(3,4,11,7)(5,8,14,16)(10,12,13,15)(33,34,40,46)(35,44,38,41)(36,43,37,42)(39,45,48,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,12)(10,14)(15,16)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(33,35)(34,41)(36,45)(37,47)(38,40)(39,43)(42,48)(44,46);(5,14)(8,16)(10,13)(12,15)(18,23)(20,29)(24,30)(26,32)(33,36,40,37)(34,42,46,43)(35,45,38,47)(39,41,48,44);(1,4,6,7)(2,3,9,11)(5,12,14,15)(8,10,16,13)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,37,40,36)(34,43,46,42)(35,45,38,47)(39,41,48,44);(1,5,6,14)(2,8,9,16)(3,10,11,13)(4,12,7,15)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,38,40,35)(34,44,46,41)(36,45,37,47)(39,42,48,43);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,14)(15,16)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,39)(34,45)(35,43)(36,41)(37,44)(38,42)(40,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=gi,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, 8525190);