#508134

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18,21,26)(19,29,31,23)(20,25,34,28)(22,24,36,41)(27,43,42,35)(30,39,33,38)(32,47,46,40)(37,45,44,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,31)(18,23,26,29)(20,30,34,33)(22,32,36,46)(24,40,41,47)(25,38,28,39)(27,37,42,44)(35,45,43,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(18,24)(19,30)(20,34)(22,35)(23,37)(25,42)(26,41)(27,28)(29,44)(31,33)(32,46)(36,43)(38,40)(39,47);(4,12)(7,14)(10,16)(13,15)(17,20,21,34)(18,25,26,28)(19,30,31,33)(22,35,36,43)(23,38,29,39)(24,27,41,42)(32,45,46,48)(37,47,44,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415330);