#505197

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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	#25777	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

755 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)(15,29)(16,33)(17,37)(18,26)(19,25)(20,40)(21,28)(22,30)(23,31)(24,36)(27,39)(32,44)(34,43)(35,42)(38,41);(1,2)(3,4)(6,9)(10,12)(13,15,19,16,20,31,34,35,21,32,38,36,39,17,18,30)(14,22,26,37,27,24,41,44,28,42,43,23,40,33,25,29);(2,4)(5,7,8,11)(6,10,9,12)(13,16,34,32,39,30,19,31,21,36,18,15,20,35,38,17)(14,23,41,22,40,44,26,33,28,37,25,42,27,29,43,24);(6,9)(10,12)(13,17,38,35,20,15,18,36,21,31,19,30,39,32,34,16)(14,24,43,29,27,42,25,37,28,33,26,44,40,22,41,23)

Polycyclic presentation:

2^9|b^2=f,c^2=ei,d^2=fi,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg

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, 412393);