#8893722

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	#57181	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,f^2=g,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, 8800918);