#506247

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

668 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413443);