#508944

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fh,h^c=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, 416140);