#466683

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

830 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)(5,6,8,9)(7,10,11,12)(13,14)(15,21)(16,22)(17,33)(18,24)(19,35)(20,25)(23,34)(26,36)(27,42)(28,37)(29,43)(30,38)(31,39)(32,40)(41,44);(6,9)(10,12)(13,15,18,28,20,30,19,29)(14,21,24,37,25,38,35,43)(16,31,32,41,17,27,34,26)(22,39,40,44,33,42,23,36);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,16,20,17)(14,22,25,33)(15,26,30,41)(18,34,19,32)(21,36,38,44)(23,35,40,24)(27,29,31,28)(37,42,43,39);(1,3,4,2)(7,11)(10,12)(13,17,20,16)(14,23,25,40)(15,27,30,31)(18,34,19,32)(21,36,38,44)(22,35,33,24)(26,29,41,28)(37,39,43,42)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fi,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=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, 373879);