#441152

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

1408 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:

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

Polycyclic presentation:

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

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, 348348);