#444273

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1158 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)(5,6,8,10)(7,11,12,9)(13,14)(15,21)(16,17)(18,24)(19,25)(20,35)(22,31)(23,33)(26,42)(27,30)(28,38)(29,39)(32,36)(34,40)(37,43)(41,44);(2,3)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,30)(17,27)(18,28)(19,34)(20,29)(22,41)(23,26)(24,38)(25,40)(31,44)(32,43)(33,42)(35,39)(36,37);(6,10)(9,11)(13,16,20,32,18,31,19,23)(14,22,35,33,24,17,25,36)(15,26,34,44,28,43,29,30)(21,37,40,27,38,42,39,41);(1,3,4,2)(7,12)(9,11)(13,17,18,22)(14,23,24,32)(15,27,28,41)(16,25,31,35)(19,33,20,36)(21,26,38,43)(29,37,34,42)(30,40,44,39)

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=gi,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,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, 351469);