#504706

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

563 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)(7,12)(8,10)(9,11)(13,14,21,28)(15,23,30,42)(16,22,33,41)(17,34,32,39)(18,26,35,44)(19,25,36,38)(20,37,31,27)(24,43,40,29);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,29,20,17,35,33,21,30,36,43,31,32,18,16)(14,22,26,39,27,40,38,42,28,41,44,34,37,24,25,23);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,32,31,43,36,30,21,33,35,17,20,29,19,15)(14,23,25,24,37,34,44,41,28,42,38,40,27,39,26,22);(6,11)(10,12)(13,17,19,33,20,30,35,43,21,32,36,16,31,15,18,29)(14,24,26,23,27,22,38,39,28,40,44,42,37,41,25,34)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=e,d^2=f,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, 411902);