#511983

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

985 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

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, 419179);