#455903

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

991 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:

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

Polycyclic presentation:

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

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, 363099);