#8900043

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

3843 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,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, 8807239);