#433264

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1075 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)(6,10)(11,12)(13,14,19,25,18,24,20,37)(15,26,29,44,28,21,30,40)(16,31,33,39,27,42,36,22)(17,34,32,23,35,41,38,43);(2,3)(5,6)(7,11)(8,12)(9,10)(13,15,19,29,18,28,20,30)(14,21,25,40,24,26,37,44)(16,32,33,35,27,38,36,17)(22,41,31,43,39,34,42,23);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,22)(15,17)(18,27)(19,36)(20,33)(21,23)(24,39)(25,42)(26,43)(28,35)(29,38)(30,32)(31,37)(34,40)(41,44);(1,3,4,2)(5,8,9,7)(6,11,10,12)(13,17,18,35)(14,23,24,43)(15,27,28,16)(19,32,20,38)(21,39,26,22)(25,41,37,34)(29,36,30,33)(31,40,42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=f,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=de,f^c=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340460);