#502400

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	#28225	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

611 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)(7,11,12,10)(13,14,16,22,32,41,33,40,19,25,17,23,18,24,31,21)(15,20,28,36,44,34,26,42,27,35,29,37,30,38,43,39);(1,2)(3,4)(6,9)(10,11)(13,15,16,28,32,44,33,26,19,27,17,29,18,30,31,43)(14,20,22,36,41,34,40,42,25,35,23,37,24,38,21,39);(2,4)(14,21)(15,26)(16,31)(17,33)(18,32)(20,34)(22,24)(23,41)(25,40)(27,43)(28,44)(29,30)(35,38)(39,42);(5,7,8,12)(6,10,9,11)(15,27)(20,35)(26,43)(28,29)(30,44)(34,38)(36,37)(39,42)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409596);