#433414

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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

1006 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)(7,9)(8,10)(11,12)(13,14,20,26,18,24,19,25)(15,27,38,40,29,21,30,41)(16,31,34,22,28,44,33,43)(17,35,37,23,36,42,32,39);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,20,38,18,29,19,30)(14,21,26,41,24,27,25,40)(16,32,34,17,28,37,33,36)(22,42,44,39,43,35,31,23);(1,2,4,3)(6,10)(9,12)(13,16)(14,22)(15,17)(18,28)(19,34)(20,33)(21,39)(23,27)(24,43)(25,44)(26,31)(29,36)(30,37)(32,38)(35,40)(41,42);(1,3,4,2)(13,17,18,36)(14,23,24,39)(15,28,29,16)(19,32,20,37)(21,22,27,43)(25,35,26,42)(30,34,38,33)(31,41,44,40)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ei,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,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, 340610);