#498213

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

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

588 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)(15,24)(16,22)(17,20)(18,30)(19,33)(21,27)(23,32)(25,41)(26,40)(28,36)(29,43)(31,37)(34,39)(35,42)(38,44);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(14,20)(21,35)(22,33)(23,36)(24,40)(30,43)(37,38)(39,41);(5,7,8,11)(6,10,9,12)(13,15)(14,21)(16,25)(17,26)(18,31)(19,34)(20,35)(22,37)(23,39)(24,30)(27,32)(28,42)(29,44)(33,38)(36,41)(40,43);(6,9)(10,12)(13,16,17,19)(14,22,20,33)(15,25,26,34)(18,28,29,32)(21,37,35,38)(23,30,36,43)(24,41,40,39)(27,31,42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=dei,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, 405409);