#409044

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1189 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fhi,c^2=f,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi

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, 316240);