#8813954

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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721150);