#504693

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

566 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)(13,14,20,27)(15,23,32,43)(16,34,30,22)(17,36,33,41)(18,38,29,26)(19,39,37,25)(21,40,35,28)(24,44,42,31);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,31,21,33,18,30,20,32,37,44,35,17,29,16)(14,22,26,41,28,24,25,23,27,34,38,36,40,42,39,43);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,33,35,44,19,15,20,30,29,17,21,31,37,32)(14,23,25,42,40,41,26,34,27,43,39,24,28,36,38,22);(6,11)(10,12)(13,17,37,30,21,15,29,44,20,33,19,16,35,32,18,31)(14,24,38,43,28,34,39,41,27,42,26,23,40,22,25,36)

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,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^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, 411889);