#122209

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₁₆

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₈²	C₂³
commutator, socle	C₂²	C₂×C₈²
prime radical	C₂³	C₈²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,c^a=ci,c^b=ch

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226965, #72226969, #72226971, …
maximal normal subgroup of	#72226965, #72226969, #72226971, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29405);