#498219

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

596 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=dei,d^c=di,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, 405415);