#8900048

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8807244);