#500906

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408102);