#10218706

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	#45994	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4411 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=g,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, 10125902);