#508440

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

555 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, 415636);