#458695

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
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

1033 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)(3,4)(5,6,9,11)(7,10,8,12)(13,14,17,22,18,23,33,38)(15,24,27,36,28,41,31,19)(16,20,29,39,30,40,32,21)(25,42,43,37,26,35,44,34);(5,7,9,8)(6,10,11,12)(13,15,18,28)(14,19,23,36)(16,26,30,25)(17,31,33,27)(20,37,40,34)(21,35,39,42)(22,41,38,24)(29,43,32,44);(5,8,9,7)(6,10,11,12)(13,16,17,29,18,30,33,32)(14,20,22,39,23,40,38,21)(15,25,27,43,28,26,31,44)(19,34,24,42,36,37,41,35);(2,4)(13,16)(14,21)(15,26)(17,32)(18,30)(19,35)(20,38)(22,40)(23,39)(24,37)(25,28)(27,43)(29,33)(31,44)(34,41)(36,42)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365891);