#497673

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

838 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)(6,10)(11,12)(13,14)(15,24)(16,21)(17,20)(18,31)(19,25)(22,30)(23,29)(26,32)(27,34)(28,36)(33,42)(35,40)(37,44)(38,43)(39,41);(5,6,9,10)(7,11,8,12)(13,15,16,25)(14,19,21,24)(17,26,28,40)(18,27,29,41)(20,35,36,32)(22,33,37,38)(23,34,31,39)(30,43,44,42);(5,7,9,8)(6,11,10,12)(14,20)(15,25)(17,30)(18,29)(19,32)(21,36)(22,23)(24,35)(26,42)(28,44)(31,37)(33,39)(34,38)(40,43);(5,8,9,7)(6,11,10,12)(15,25)(19,24)(26,40)(27,41)(32,35)(33,38)(34,39)(42,43)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ei,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404869);