#8932819

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

3982 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)(3,8,11,14)(4,13,15,9)(5,12,16,10)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(35,41)(36,44)(39,42)(40,45);(2,7)(3,11)(9,13)(10,12)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,34)(35,42)(36,45)(37,38)(39,41)(40,44)(43,47)(46,48);(2,7)(5,16)(8,14)(9,13)(21,29)(24,28)(26,32)(30,31)(34,38)(39,42)(40,45)(47,48);(1,3,6,11)(2,8,7,14)(4,10,15,12)(5,9,16,13)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,35)(34,39)(36,46)(37,41)(38,42)(40,48)(43,44)(45,47);(1,4,6,15)(2,9,7,13)(3,12,11,10)(5,14,16,8)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,47)(41,46)(42,48);(1,5)(2,10)(3,9)(4,14)(6,16)(7,12)(8,15)(11,13)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840015);