#441870

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fi,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 349066);