#420593

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1049 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,d^2=ei,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^b=gi,g^c=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, 327789);