#433268

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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

1065 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)(13,14,20,26,18,24,19,25)(15,27,31,40,29,21,30,41)(16,32,36,22,34,43,35,39)(17,37,38,23,28,42,33,44);(2,3)(5,6)(7,11)(8,12)(9,10)(13,15,19,30,18,29,20,31)(14,21,25,40,24,27,26,41)(16,33,35,28,34,38,36,17)(22,42,32,23,39,37,43,44);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,22)(15,28)(17,29)(18,34)(19,36)(20,35)(21,23)(24,39)(25,43)(26,32)(27,44)(30,33)(31,38)(37,41)(40,42);(1,3,4,2)(5,8,9,7)(6,11,10,12)(13,17,18,28)(14,23,24,44)(15,16,29,34)(19,33,20,38)(21,39,27,22)(25,37,26,42)(30,35,31,36)(32,40,43,41)

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=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, 340464);