#421047

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1852 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=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, 328243);