#397005

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

1046 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)(7,11)(8,12)(9,10)(13,14)(15,25)(16,20)(17,22)(18,23)(19,27)(21,29)(24,34)(26,42)(28,40)(30,37)(31,38)(32,39)(33,41)(35,43)(36,44);(1,2,4,3)(6,10)(11,12)(13,15)(14,19)(16,30)(17,27)(18,29)(20,36)(21,24)(22,25)(23,40)(26,32)(28,34)(31,44)(33,43)(35,39)(37,38)(41,42);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,17,31)(14,20,22,38)(15,26,27,43)(18,33,34,32)(19,35,25,42)(21,37,40,36)(23,41,24,39)(28,44,29,30);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,15)(14,21)(16,26)(17,27)(18,28)(19,24)(20,37)(22,40)(23,25)(29,34)(30,32)(31,43)(33,44)(35,39)(36,38)(41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304201);