#421062

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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	C₂×#3013	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1872 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)(15,26)(16,30)(17,36)(18,25)(19,24)(20,23)(21,28)(22,34)(27,42)(29,38)(31,37)(32,39)(33,40)(35,41)(43,44);(1,2,3,4)(5,7,8,11)(6,9,10,12)(13,15,18,28,19,17,20,29)(14,21,23,26,24,38,25,36)(16,31,33,27,34,43,35,32)(22,39,40,44,30,42,41,37);(6,10)(9,12)(13,16,19,34)(14,22,24,30)(15,27,17,32)(18,33,20,35)(21,37,38,44)(23,40,25,41)(26,39,36,42)(28,43,29,31);(2,4)(13,17)(14,21)(15,19)(16,32)(18,28)(20,29)(22,37)(23,36)(24,38)(25,26)(27,34)(30,44)(31,35)(33,43)(39,41)(40,42)

Polycyclic presentation:

2^9|b^2=efi,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,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, 328258);