#504699

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

555 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)(15,29,33,44)(16,24,36,43)(17,37,34,23)(18,39,31,26)(19,40,30,25)(21,41,35,28)(22,38,42,32);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,30,36,21,34,31,38,20,33,19,16,35,17,18,32)(14,22,39,37,28,24,40,44,27,42,26,23,41,43,25,29);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,33,35,38,19,34,20,36,31,15,21,32,30,17)(14,23,25,42,41,44,26,24,27,37,40,22,28,29,39,43);(6,11)(10,12)(13,17,30,32,21,15,31,36,20,34,19,38,35,33,18,16)(14,24,39,44,28,42,40,23,27,43,26,29,41,22,25,37)

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

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, 411895);