#8416763

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	#57384	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8323959);