#9053924

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3784 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)(3,8)(4,14)(5,9)(6,15)(7,10)(11,12)(13,16)(17,18,23,24)(19,27,28,25)(20,30,29,26)(21,22,31,32)(34,38)(36,43)(39,44)(42,47);(1,3,7,13)(2,8,10,16)(4,9,15,11)(5,6,12,14)(18,24)(19,28)(21,31)(26,30)(35,41)(39,44)(42,47)(46,48);(1,4,7,15)(2,6,10,14)(3,11,13,9)(5,16,12,8)(17,19,23,28)(18,25,24,27)(20,21,29,31)(22,30,32,26)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4)(2,6)(3,9)(5,16)(7,15)(8,12)(10,14)(11,13)(17,20,23,29)(18,26,24,30)(19,21,28,31)(22,27,32,25);(1,5,7,12)(2,9,10,11)(3,6,13,14)(4,8,15,16)(17,21)(18,22)(19,29)(20,28)(23,31)(24,32)(25,30)(26,27)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,6)(2,4)(3,12)(5,13)(7,14)(8,9)(10,15)(11,16)(17,22)(18,21)(19,30)(20,27)(23,32)(24,31)(25,29)(26,28)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961120);