#9009208

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

3992 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8916404);