#497672

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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

806 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)(17,29)(18,33)(19,23)(20,27)(22,31)(25,40)(26,41)(28,39)(30,36)(32,43)(34,38)(35,42)(37,44);(5,6,9,10)(7,11,8,12)(15,25)(20,35)(24,40)(26,34)(27,42)(30,44)(36,37)(38,41);(5,7,9,8)(6,11,10,12)(13,15)(14,20)(16,26)(17,30)(18,34)(19,25)(21,36)(22,38)(23,35)(24,29)(27,31)(28,42)(32,44)(33,37)(39,41)(40,43);(5,8,9,7)(6,11,10,12)(13,16,19,18)(14,21,23,33)(15,26,25,34)(17,28,32,31)(20,36,35,37)(22,29,39,43)(24,41,40,38)(27,30,42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,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, 404868);