#8893416

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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)(3,7,10,14)(4,5,8,13)(11,12,15,16)(17,18)(19,28)(20,31)(21,32)(22,23)(24,27)(25,30)(26,29)(33,34,38,42)(35,43,44,39)(36,40,45,48)(37,41,46,47);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,19,24,30)(18,25,27,28)(20,29,22,21)(23,26,31,32)(35,44)(37,46)(39,43)(41,47);(3,10)(7,14)(11,15)(12,16)(17,20,24,22)(18,23,27,31)(19,29,30,21)(25,26,28,32)(33,35,38,44)(34,39,42,43)(36,37,45,46)(40,47,48,41);(1,4)(2,5)(3,11)(6,8)(7,12)(9,13)(10,15)(14,16)(17,21)(18,26)(19,20)(22,30)(23,28)(24,29)(25,31)(27,32)(33,36)(34,40)(35,37)(38,45)(39,47)(41,43)(42,48)(44,46);(1,5)(2,8)(3,12)(4,9)(6,13)(7,15)(10,16)(11,14)(17,22,24,20)(18,23,27,31)(19,29,30,21)(25,32,28,26)(33,37)(34,41)(35,45)(36,44)(38,46)(39,40)(42,47)(43,48);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,23)(18,20)(19,26)(21,28)(22,27)(24,31)(25,29)(30,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=g,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, 8800612);