#508958

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

573 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)(21,22,25,31,24,30,38,49,27,33,41,50,26,32,40,42)(23,28,35,44,37,48,52,47,36,45,51,46,34,43,39,29);(1,2,3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23)(22,28)(24,37)(25,35)(26,34)(27,36)(29,42)(30,48)(31,44)(32,43)(33,45)(38,52)(39,40)(41,51)(46,50)(47,49);(2,4)(8,16)(11,18)(14,20)(17,19)(21,23)(22,29)(24,34)(25,39)(26,37)(27,36)(28,42)(30,46)(31,43)(32,44)(33,47)(35,40)(38,51)(41,52)(45,49)(48,50);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,27,26)(22,30,33,32)(23,34,36,37)(25,38,41,40)(28,43,45,48)(29,46,47,44)(31,49,50,42)(35,39,51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416154);