#508123

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,26,31)(23,27,32,40)(24,28,37,44)(25,38,36,30)(29,45,43,35)(33,39,46,50)(34,49,47,42)(41,52,51,48);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23)(22,27)(24,33)(25,34)(26,32)(28,39)(29,41)(30,42)(31,40)(35,48)(36,47)(37,46)(38,49)(43,51)(44,50)(45,52);(2,4)(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(22,28)(24,35)(25,36)(27,39)(29,30)(31,44)(33,48)(34,47)(37,45)(38,43)(40,50)(41,42)(46,52)(49,51);(8,16)(11,18)(14,20)(17,19)(23,32)(27,40)(33,46)(34,47)(39,50)(41,51)(42,49)(48,52)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415319);