#10504475

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	#53658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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)(3,10,4,7)(5,13,14,8)(11,15,12,16)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(18,24)(20,25)(26,31)(28,32)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,19,23,21)(18,20,24,25)(22,27,30,29)(26,28,31,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,20,23,25)(18,19,24,21)(22,28,30,32)(26,27,31,29)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21,23,19)(18,25,24,20)(22,27,30,29)(26,28,31,32)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

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, 10411671);