#504773

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

607 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411969);