#8917848

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	#59970	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3996 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)(3,9,10,4)(5,13,14,8)(11,12,16,15)(33,34)(35,39)(36,40)(37,44)(38,42)(41,48)(43,45)(46,47);(2,7)(4,9)(8,13)(12,15)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28)(33,35,38,45)(34,39,42,43)(36,41,47,44)(37,40,48,46);(2,7)(3,10)(5,14)(12,15)(18,23)(19,26)(21,30)(27,29)(33,34,38,42)(35,43,45,39)(36,40,47,46)(37,44,48,41);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19)(18,24)(20,29)(21,28)(22,26)(23,31)(25,27)(30,32)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,45)(44,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28)(33,36,38,47)(34,40,42,46)(35,44,45,41)(37,43,48,39);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)(33,37)(34,41)(35,40)(36,43)(38,48)(39,47)(42,44)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8825044);