#8957568

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	#49405	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(3,9)(4,8)(5,11)(6,7)(10,16)(12,14)(13,15)(17,18,23,20)(19,24,26,22)(21,25,29,28)(27,31,32,30)(35,38)(40,42)(43,45)(47,48);(1,3)(2,5)(4,12)(6,11)(7,9)(8,16)(10,13)(14,15)(17,19,23,26)(18,24,20,22)(21,27,29,32)(25,31,28,30)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(2,7)(4,13)(5,9)(10,12)(19,26)(21,29)(22,24)(25,28)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3)(2,5)(4,12)(6,11)(7,9)(8,16)(10,13)(14,15)(17,20,23,18)(19,22,26,24)(21,25,29,28)(27,31,32,30)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,4,6,13)(2,8,7,15)(3,10,11,12)(5,14,9,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,5)(2,3)(4,14)(6,9)(7,11)(8,10)(12,15)(13,16)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,32)(29,31)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8864764);