#505223

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

704 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)(13,14,21,28)(15,23,17,33)(16,22,32,24)(18,26,37,42)(19,25,34,44)(20,38,36,27)(29,41,35,40)(30,43,31,39);(1,2)(3,4)(6,9)(10,12)(13,15,19,29,20,30,37,32,21,17,34,35,36,31,18,16)(14,22,26,39,27,40,44,33,28,24,42,43,38,41,25,23);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,31,36,35,34,17,21,32,37,30,20,29,19,15)(14,23,25,41,38,43,42,24,28,33,44,40,27,39,26,22);(6,9)(10,12)(13,17,34,29,20,31,18,32,21,15,19,35,36,30,37,16)(14,24,42,39,27,41,25,33,28,22,26,43,38,40,44,23)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,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,d^b=di,d^c=di

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, 412419);