#504358

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411554);