#8901423

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	#47245	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=hi,e^2=hi,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, 8808619);