#443787

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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

900 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,25,30,24)(15,22,17,33,28,39,27,20)(16,21,31,36,32,38,40,37)(26,35,29,43,42,44,41,34);(2,3)(5,7,8,11)(6,9,10,12)(13,15,19,28)(14,20,25,33)(16,29,32,41)(17,18,27,30)(21,35,38,44)(22,23,39,24)(26,40,42,31)(34,37,43,36);(6,10)(9,12)(13,16,30,40,19,32,18,31)(14,21,24,37,25,38,23,36)(15,26,27,41,28,42,17,29)(20,34,39,44,33,43,22,35);(1,3,4,2)(13,17,18,28,19,27,30,15)(14,22,23,33,25,39,24,20)(16,29,31,42,32,41,40,26)(21,35,36,43,38,44,37,34)

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=gi,d^2=e,e^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,e^b=ei,g^b=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350983);