#10219519

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	#51904	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4405 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10126715);