#508799

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,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, 415995);