#8902466

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	#57810	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=hi,e^2=hi,f^2=gh,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, 8809662);