#438942

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

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

796 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)(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,20)(17,22)(18,24)(21,31)(23,25)(26,33)(27,37)(28,34)(29,36)(30,38)(32,39)(35,40)(41,44)(42,43);(2,3)(13,15,25,40,18,29,17,28)(14,19,23,35,24,36,22,34)(16,27,31,26,32,41,30,42)(20,37,21,33,39,44,38,43);(1,2,4,3)(5,7,8,12)(6,9,10,11)(13,16)(14,20)(15,26)(17,31)(18,32)(19,33)(21,22)(23,38)(24,39)(25,30)(27,40)(28,41)(29,42)(34,44)(35,37)(36,43);(1,3,4,2)(13,16)(14,21)(15,27)(17,30)(18,32)(19,33)(20,22)(23,39)(24,38)(25,31)(26,40)(28,42)(29,41)(34,37)(35,44)(36,43)

Polycyclic presentation:

2^9|a^2=ef,b^2=egi,c^2=ef,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,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, 346138);