#504707

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

542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,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, 411903);